Semigroups of operators, cosine operator functions, and linear differential equations

SEMIGROUPS OF OPERATORS, COSINE OPERATOR FUNCTIONS, AND LINEAR DIFFERENTIAL EQUATIONS V. V. Vasil'ev, S. G. Krein, and ...

Author: Krein S. | Khazan

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SEMIGROUPS OF OPERATORS, COSINE OPERATOR FUNCTIONS, AND LINEAR DIFFERENTIAL EQUATIONS

V. V. Vasil'ev, S. G. Krein, and S. I. Piskarev

UDC 517.986.7+517.983.5

This survey presents a systematic exposition of the elements of the theory of operator semigroups (OS's) in Banach space from Hille--Yosida to the end of 1989. There is a parallel exposition of the theory of cosine operator functions (COF's). The paper contains the following divisions." Linear differential equations in Banach space, reduction of the Cauchy problem for second order equations to the Cauchy problem for first order equations, one-parameter OS's and COF's. differenHable OS's. analytic OS's, Fredholm OS's. positive OS's. stable OS's. spectral properties of OS's and COF's. compactness properties of OS's and COF's, uniformly continuous OS's and COF's. almost periodic OS'.s and COF's. uniformly bounded OS's and COF's. the theory of perturbalion.~ for OS's and COF'.s, adjoint OS's and COF's. INTRODUCTION Individual results obtained prior to 1983 in the theory of linear and nonlinear semigroups of operators and its applications were surveyed by S. G. Krein and M. 1. Khazan in [60]. In that survey the authors announced a proposal for a second part that would consider other classes of differential equations in Banach spaces. However, in studying the literature that appeared between 1983 and 1989, it was discovered that there was a large group of publications on various problems in the theory of linear semigroups; of these publications, several monographs were particularly illuminating: [126, 162, 211,213, 229, 341,390]. Unfortunately, practically nothing on these problems was published in Russian. As a result, a large part of the present survey is devoted to presenting anew the theory of linear semigroups of the class C o (OS's). For continuity in exposition we sometimes restate results that have already been presented in [60]. Another object of study in the present paper is cosine operator functions (COF's) associated with second-order differential equations of hyperbolic type in Banach space, the theory of which has developed vigorously in the past twenty years. There is, for example, the monograph [213], which is devoted entirely to this theory. In Russian there are neither monographs nor surveys. It should be noted that the theory of COF's runs parallel to the theory of semigroups. Thus, in each chapter of the present survey we first present material on semigroup theory, and then the corresponding COF theory. In particular, this shows how the two complement each other. A priori constraints on the size of the survey made it impossible to include classical problems such as, for example, application of fractional powers and interpolation spaces, the theory of boundary-value problems for equations of second and higher order in Banach space, asymptotic methods for investigating equations with a small parameter. coercivity inequalities, problems on discrete approximation of differential equations, or many applications. In a word. our intention is to provide a springboard for discussion of the modern theory of differential equations in Banach space, and we assume that these problems will be considered later. To conclude our advertisement for the theory of OS's and COF's, we are deeply gratified to note that the monograph [229] is being published by "Vysshaya Shkola," and Mir will be publishing a translation of [178] in 1991. The paper is structured as follows: There are 9 sections, each of which contains divisions that are further divided into paragraphs. A paragraph contains a complete proposition. Accordingly, a tripartite numbering system is used: for example, 1.3.2 is the second proposition of the third division of the first section. Within each section, the section numbers are omitted. Thus, formulas within a section have only two-position numbering, although references from out-

Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz. Vol. 28. pp. 87-202. 1990. 1042

0090-4104/91/5404-1042S12.50 9

Plenum Publishing Corporation

side the section in which the formula is defined contain the section number. Bibliographic references are not necessarily primary, since we frequently refer to publications in which a given problem is discussed in sufficient detail. The bulk of our notation is in c o m m o n use. For the reader's convenience the survey contains a glossary of notation. Equations used to define some quantity are indicated with a := sign. The authors would like to thank V. I. K o n o n e n k o for help in selecting material. 1. L I N E A R D I F F E R E N T I A L E Q U A T I O N S IN B A N A C H S P A C E 1.1. Preliminary Remarks. Let E be a Banach space over the field of complex numbers with norm ]1" II. We denote the set of linear operators mapping D(A) C E into E by L(E), and the continuous linear operators by B(E). We denote the domain and range of an operator A by D(A) and ~ ( A ) , respectively. Equipped with the norm [llxlll :=[Ixll+llAxl[ , the linear manifold D(A) becomes a Banach space, which we denote by ~ (A)+ We use traditional notation for the resolvent set p(A) and spectrum a(A) of the operator A. We denote by (D(A) ~ E ) the set containing the operators that are closed in E, have dense domains ~ ( E ) ~ L ( E ) , and have nonempty resolvent sets (p(A) ~ O). Let J C R := ( - ~ , + ~ ) and N = {1,2,...}. Definition 1. A f u n c t i o n u(t) : J ---, E is said to be continuous at a point t o ~ J if

lira Ji u(t)--u(to)ll = 0 .

t+to

A functions is said to be continuous on J if it is continuous at every point t ~ J; in this case we write u(t) ~ C(J;E). Definition 2. The derivative o f a function u(t) at a point t o E J is an element u'(t o) E E such that lira fj {u(to +

At)--tt(to))/ At -- it'(to)~]----0.

At~O

If a function u(t) has a continuous derivative u'(t) at every point t ~ J, it is said to be continuously differentiable on J. In this case we write ,u~ Ci(J;E). The definition o f the set of k-times continuously differentiable functions c k ( j ; E ) is analogous to Def. 2. 1.2. C a u c h y Problem for n - t h - o r d e r Differential Equations. In a Banach space E, we consider the Cauchy problem for an n - t h order equation of the f o r m

u
/(~R+:-----[0, oo),

u(0) = u ~ u'(0) = u I. . . . . where

(2.1)

u+"-" (0) ----u"-I,

A 6 ~ ( E ) , and u(n)(t) denotes the n - t h derivative (i.e., (d/dt)nu(t)) o f the function u(t). Definition 1. A function u(t) : R+ ~ E is said to be a solution of Eq. (2.1) if u(t)~C"(R+;

(2.2)

E), u(t)~D(A)

for

any t E R+ and (2.1) is satisfied. Definition 2. A C a u c h y problem (2.1)-(2.2) is said to be u n i f o r m l y well-posed on l~+ if: (i) there exists a dense subset D c E such that for u~ x..... u(n-l) E D there exists a unique solution on R+ of Eq. (2.1) with the p r o p e r t y that u~~) (0): = lira tt (~) (h)-----ttk; k --=O, r t - - l "

(i.e., k runs through all integers f r o m 0 to n - - 1);

h~O

u~-+O

(ii) the solution o f (2.1)-(2.2) is u n i f o r m l y stable with respect to t on any c o m p a c t ,7g'cl(+ , i.e., the conditions (u~ED; k -----0,n----~) as p ~ oo, imply convergence o f the solutions Up(t) -+ 0 o f (2.1)-(2.2) u n i f o r m l y with ~k)

~.

respect to tEYd , where up (O)=u o, k----O, n - Y . To study the properties o f solutions to uniformly well-posed C a u c h y problem (2.1)-(2.2) we introduce the propagators ,Sty(t), ] = 0 , n - - I , which provide a solution of (2.1)-(2.2) with initial conditions uj(k)(0) = 6jkX (6ik is the K r o n e c k e r delta), i.e., uj(t) =,,ar~(t)x. 1.2.1. [209, 381] The operators

~r~lt), ! = 0 , n - - l " are densely defined, strongly continuous in t, and for any

t may be extended continuously to all of E.

1043

1.2.2. [209, 381] If u(t) is a solution of (2.1)-(2.2), then

n-1

.(t)=~ ~rj(t)uJ,

(-..~)

tef~+.

2=0

T H E O R E M 1 ([209, 53]). The C a u c h y problem (2.1)=(2.2) with n >_ 3 is u n i f o r m l y welt-posed if and only if A B(E), i.e., A is a continuous operator with D(A) = E. In this case we have

Y'/(t)= ~

lkn+/A~/(nk-b j )L

(2.4)

k=0

1.3. Cauchy Problem for First Order Equations and Co-semigroups. In order to consider the case n = 1, we present the following definition. Definition 1. A o n e - p a r a m e t e r family of operators U(t) ~ B(E), t E R+, is said to be a semigroup of operators on E if

(i)U(t+s)=U(t)U(s) (ii) U ( 0 ) = I A semigroup is said to be strongly continuous if lim

for all

is the identity

t, S~0;

operator

on E.

U(l)x -~-U(s)x (s > 0, x6E).

(3,i)

Finally, a semigroup belongs to the

t-.*.S

class C o (and we will call it an OS) if (iii) limU(t)x=x-~ for all x ~ E. t~0+

Definition 2. A linear operator A with domain

D(A):={xEE: lira ((U(h)--I)/h)x exists

}

(3.2)

xED(A)

(3.3)

h--0+

and defined so that

Ax:=lim((U(h)--I)/h)x h-*O+

for

is called the generating operator (infinitesimal generator) o f the OS U(t). In the sequel we will usually denote a semigroup with generating operator A by exp(tA). 1.3.1. [288] The generating operator of an OS belongs to ~ (E).. T H E O R E M 1 ([59]). A C a u c h y problem (2.1)-(2.2) with n = 1 is uniformly well-posed if and only if the operator A is the generating o p e r a t e / o f an OS. 1.3,2. The solution o f a u n i f o r m l y well-posed problem (2.1)-(2.2) is given by the f o r m u l a

u(t) = e x p (tA)u ~ t~O.

(3.4)

1.4. Cauchy Problem for Second Order Differential Equations and Cosine Operator Functions. Definition 1. Let ~ be a Banach algebra with identity e-. A mapping C(t): R~-~-.~ is said to be a cosine function if (i) C(t + s) + C(t - s) = 2C(t)C(s) for all t, s E R (the d'Alembcrt equation): (ii) C(0) = "e.

1044

When 9 ~ = B ( E )

, where E is a Banach space, ~ = 1 (the identity operator), it is usually required that

(iii) the operator function C(t) is strongly continuous in E. In what follows we will be interested primarily in cosine operator functions with properties (i)-(iii), which we will call COF. A COF C(t) is associated with a sine operator function (SOF)

t

S(t)x:-----I C(s)xds,

tER, xfiE

(4.2)

0

and linear manifolds E k : ={xEE :

C(t)xECk(R; E)}; k = l , 2.

Definition 2. A linear operator A with domain D(A) that consists of all x for which the limit

Iim ((C ( 2 h ) - -

h~O

I)/2k)/x: = Ax

(4.3)

exists is said to be the (infinitesimal) generating operator for the COF C(t). 1.4.1. [381] The generating operator for a COF is densly defined, closed, and belongs to W ( E ) . We denote the COF and SOF with generating operator A by C(t,A) and S(t,A), respectively. 1.4.2. [396] The domain D(A) of the generating operator A for the COF C(t,A) coincides with E z. T H E O R E M 1 [210]. A Cauchy problem (2.1)-(2.2) with n = 2 is uniformly well-posed if and only if the operator A is a generating operator for a COF. 1.4.3. [210] A solution of a uniformly well-posed problem (2.1)-(2.2) with n = 2 for all u ~ E D(A) and u 1 ~ E 1 exists and may be written in terms of a COF and an SOF as follows:

u(t) =C(t, A)uO+S(t, A)u ~, t~R.

(4.4)

1.5. Representation of Cosine Operator Functions in Terms of Groups of Operators. Assume we are given a oneparameter group of operators exp(tA) (for the definition, see 2.2) with values in a Banach algebra with identity ~. Then the function

C(t) : = (exp(tA)+exp(--tA) ) /2, t~R,

(5.1)

has the properties (4.1).

Definition 1. When 9~=B(E), is a cosine function for which representation (5.1) is possible, we say that it is an exponential cosine operator function. 1.5.1. [398] The generating operator B for an exponential COF of the form (5.t) may be represented in the form B = A 2. 1.5.2. [7, 272, 273, 327,402] Examples of COF's that are not exponential have been constructed. 1.5.3. [141] If a cosine function C(t,A) with values in ~ and properties (4.1) is measurable, the Banach algebra can be extended to a Banach algebra 2 , in which C(t,A) is exponential. 1.5.4. [141] If, for some t E R, the spectrum of a COF C(t,A) is contained in a singly connected open subset of the complex plane that does not contain unity, then C(t,A) is an exponential COF. 1.5.5. [141] It follows from 5.4 that if I[C(to,A)t[ < 1 for some to, then the COF C(t,A) is exponential. 1.5.6.[141] Let ~o b e a nilpotentelement of ~ of order 2, i.e.,~o # 0 and~o 2 = 0, and l e t a ( t , r ) : R x R---*R be a biadditive symmetric function. Then C (t, A):

= e + a ( t , t)~,, i~ a cot. 1045

1.5.7. [141] For a C O F C(t,A) to be exponential it is necessary that there exist a function I( 9 R ---, I)+ such that

ta(t, vjl..
for al'l

t,T~R+.

(5.2)

1.5.8. [190] There exists a symmetric biadditive a(t,r) such that for it there is no function I( with property (5.2). 1.5.9. [141] Thus, the study o f COF's cannot be reduced by means of representation (5.1) to the study of OS's even by extension o f the Banach algebra N . The requirement of measurability in Proposition 5.3 is essential. 1.6. S u p p l e m e n t a r y Comments. As we can see f r o m Theorems 3.1 and 4.1, u n i f o r m l y well-posed Cauchy problems are closely related to OS's and COF's. Generalization of the notion of well-posedness to the solution to a Cauchy problem leads to broader classes of semigroups and cosine operators. We note briefly possible approaches for generalization, leaving detailed discussion for a subsequent survey. Other ( n o n - s e m i g r o u p ) approaches to the study of abstract equations of the f o r m Au'(t) + Bu(t) = f(t), where A and B are self-adjoint and A _>0, may be f o u n d in [49]-[51 ]. C a u c h y problems that are well-posed in the sense of distributions, and problems associated with them, are considered in [166, 379, 404]. The publications [192, 319, 392, 393] are devoted to the study of so-called C - s e m i g r o u p s , which are used to study both first- and s e c o n d - o r d e r problems, as well as semigroups with a singularity at zero. Concerning n - p a r a m e t e r semigroups, see [178] and [241]. There are a n u m b e r o f publications (see, for example, [204,214. 216]) that deal with the so-called incomplete Cauchy problem for Eq. (2.1) on the half-axis 1/+ (i.e., part of the conditions is given at zero, and part at infinity). The solvability of such problems is related to the nature of the spectrum of the operator A that appears in (2.1) and the stability of certain semigroups. 2. S E M I G R O U P S OF O P E R A T O R S OF T H E C L A S S C o ( O S ' S )

2.1. Fundamental Properties of OS's. 2.1.1. ([288]) For any OS exp(tA) there exist constants M > 1 and co ~ R such that Ilexp(tA)II~<M exp(o~t) for

(1.1)

t~>O.

Definition 1. An OS exp(tA) is said tO be uniformly bounded if Hexp(tA)I] -< M for t >_0 and some constant M. An OS is said to be a contaction semigroup if M = 1. Note that if an OS exp(tA) satisfies bound (1.1), then e - a t exp(tA) is an OS satisfying (1.1) with w1 = co -- c~. 2.1.2. [126] For an OS exp(tA) the limit

o~(A): -----lira t -~ tn tf exp

(tA)II =

t--zo

in~ t - ' In II exp

(tA)l!

t>O

exists and is equal to inf{w ER : for every 6 > co there exists a constant M6 with the p r o p e r t y that ]]exp(tA) H -< M6 exp(6t), t _> 0).

Definition 2. The n u m b e r c0(A) is called the type of the OS exp(tA). 2.1.3. [126] The type of an OS may be either finite or equal to --oo. 2.1.4. [178] Let E = H be a Hilbert space. Then for an OS exp(tA) we have w(A) = inf{a E R : c~ + ifl E p(A) for all ~ E R and the family o f resolvents R(cz + i/3,A) with/9 ~ R is uniformly bounded). 2.1.5. [288] An operator A ~ L(E) with

D(A)=E

may be a generating operator for only one OS.

2.1.6. [35] If we are given an OS exp(tA) and Re ), > w(A), then A E p(A) and co

R(k, A)x= f e-;'t exp(tA)xdt 0

1046

for a l l

xEE.

(1.2)

The problem o f whether an operator A generates an OS is the subject of the so-called Hille--Yosida--Feller-Phillips--Miyadera Theorem. T H E O R E M 1 ([35]). Operator A ~ L(EP) is a generating operator for OS exp(tA) if and only if

AE~(E)

and

there exist constants M and co such that {A E C : Re A > aJ} C p(A) and

kl

IiR(L, A)nl!-,< (ReZ--~)" We will indicate that operator A e T (E)

for

ReL>o),

hEN.

(1.3)

generates OS exp(tA) satisfying estimate (1.1) by writing Ae,~(M, m).

2.1.7. [35] If the OS in the statement of T h e o r e m I is a contraction OS, then condition (1.3) can be replaced by {A e C, Re A > 0} c p(A) and

IIR(LA)It<~I/(Re~)

for R e ~ , > 0 .

2.1.8. [35] If some h a l f - p l a n e Re A > ~o is included in p(A) and the operator Ae~e(E) is a generating operator for an OS.

(1.4)

IIR(A,A)1]

_ (Re A -- ~o)- 1 with Re A > co, then

T H E O R E M 2 ([44], [101]). For an OS exp(tA) we have (i) if x E D(A) then the function t ---, exp(tA)x is differentiable with respect to x and

a___exp(tA)x---- A exp(tA)x=exp(tA)Ax dt

for t > O ;

!

(ii) for any x E E and t > 0 the element

f exp (sA)xds belongs to D(A) and 0

t

A fexp(sA)xds=exp(tA)x--x; 0 t

(iii) if x E D(A), then

exp(tA)x--exp(TA)x=A I exp(sA)xds

for t, r ~ 0;

(iv) the d o m a i n o f any natural power of a generating operator of an OS is invariant with respect to the semigroup, i.e., for any t ~_ 0 we have exp(tA)x ~ D(A n) for x E D(An), n E N. 2.1.9. [41] Equation (1.2) can be inverted for x ~ E and t ~ 0 in several ways: (i) (the Yosida approximation)

exp(tA)x=lim

exp(D, AR(X, A ) ) x ;

(ii) (the Widder--Post approximation formula)

2.1.10. [333] Assume that we are given an OS exp(tA) and some closed subspace F c E. The following conditions are equivalent: (i) the subspace F is invariant with respect to the operators of exp(tA) for t _> 0; (ii) there exist numbers s o E C, r ~ R, and n o E N such that F is an invariant subspace of the operators R(s o + nr,A) for all n >__n o. 2.1.11. [158] For an element x E E to be invariant with respect to an OS exp(tA) and for an OS exp(tA) to be

1047

contracting or positive, it is necessary and sufficient for the resolvent R(A,A) to have these same properties, respectively, for .~ > ~(A). t

2.1.12. [35] The set of elements of the f o r m ; exp (sA)x ds (t>O,,x(~E) in D(A) is dense in E, and, clearly, I.

t

lim t -~ f exp(sA)xds= x t ~0+

fOr

all

xEE.

(1.5)

0

2.1.13. [341] For any x ~ E, A E C, we have

t

(1.6)

(~,I - - A) l exp(), (t - - s))exp (sA}xds :-- (exp (~,t) I - - exp (tA))x. 0

Definition 3. A subset ~ D ( A ) is said to be a core of an arbitrary operator A ~ ( E ) , if the closure of its restriction A I ~ is A, or, equivalently, if . ~ is dense in D(A) in the graphic norm of A. 2.1.14. [44] If a linear manifold ~)~D(A) is dense in E and is {exp(tA),t _ 0}-invariant, then . ~ i s a c o r e of A. 2.1.15. [35, 101] For an OS exp(tA) set D(A k) is dense in E for all k E N. Moreover, set

D(A'~): =

f'l D(A k)

k=O

(sometimes called the set of infinitely differentiable vectors of A) is also dense in E and is a core of A. 2.1.16. [101] We define the set E a : = 9 ~ ( e x p ( a A ) ) for each a > 0 and /~0:= @ E a . Then Eg c_ l~a for 0 < a < 2.1.17. [126]. For OS's exp(tA) and exp(tB) the following conditions are equivalent: (i) exp(tA)exp(tB) = exp(tB)exp(tA) for t >_ 0; (ii) R(,~,A) 9 R(A,B) 9 R(A,A) for some A ~ p(A) n p(B); (iii) condition (ii) holds for all ,k ~ p(A) n p(B). When these conditions are satisfied, the following OS is defined: exp(tC) := exp(tA) x exp(tB), t ~ 1~+. The set D O := D(A) n D(B) is a core of the operator C and Cx = Ax +Bx for all x E D O9 2.1.18. [158, 333] For an OS exp(tA) we have (x ~ E) oo

exp(tA)x= lira e T M X ~ [ ~ , R ( X , ~'~-~

A)]* x;

k=O

(z7)

,?o

exp(tA)x--- lira Z ~ ( - - 1 ) ~ e 3.~a~

kx~(k_1)i I R(~,k, AJx;

k=l

exp(tA)x=re sr lim k ~ n~

a=l

( - - 1)P-le pk" •(s,+pkr, A) x (p--l)!

(18) (1.9)

uniformly on any finite segment [0,T]. 2.1.19. [158] If x ~ D(Am), then Taylor's formula m--I

exp (tA) x ~- ~ k=0

1048

1

~ Akx +

(t-- s p -1 exp (sA) Amxds

(1.10)

is valid and we have t

t

[exp(tAJ--llmx----- I " " I exp((sl + . . . + sm)A)Amxdst. . .as.,. 0

0

We define the modulus of continuity of an OS exp(tA) on [0,T] to be

r

exp(-A)x)=

sup O.J,s~T

r

}.

When T = 1, we will simply write w(6,exp(.A)x). We also set ~r : = (exp (xA) --I)/x.

(l.ll)

2.1.20. [128] For any x E E we have exp (tA)x-~ lim exp (t~r

x,

~0

where convergence is uniform on any finite segment [0,b] and

Ifexp (tA) x--

exp (t.sg~) x 1[~: r

(zt/s, exp {. AJ x) + K* I/s It x [[,

where K : =MO (Mb+4Me b' M-l,) and does not depend on r, while M := maxt~[o,b ] It exp(tA) II. 2.1.21. [158] Assume that we are given an OS exp(tA) and

A~'exp (tA): -----~=o( --1

)'-*Cmeexp((t+k*) A).

Then

1

e x p ( t A ) = [exp ( ~ ) - - 1

]mexp(tA)=a~,exp(tA )

and

exp(tA)x=lim ~

t___~_~A~exp(~A)xk=o,

where convergence is uniform with respect to t c [0,T) for all T < o~. 2.1.22. [158] Assume that we have an OS exp(tA). Then

1049

u n i f o r m l y with r e s p e c t to t E [0,1 ], w h e r e

= Cmkt~(l -- t)"-k exp

(-A-

and

{~oCmkt~(l--t)~-kexp(--~ A)}xdt--!exp(tA,xdt

FI ..<

A m o r e g e n e r a l a p p r o x i m a t i o n t h e o r e m is given in [158]. 2.1.23. [1581 A s s u m e that for t _> 0 and x E D(A m - l )

- m.

m--1

m[

tk

(1.12)

~

(the expression in p a r e n t h e s e s is the residual o f the T a y l o r series - - see Prop. 1.19). T h e n the limit Btmx exists if and only if x E D(A m) and

lira Btmx= t-.~-0+

Amx.

2.1.24. [158] A s s u m e that f o r t _> 0, and x, gi,m E E

m--I ~ tk

m!

Ptm(go,rn. . . . . gm-l,m)X:=[ff i (exp(tA)x-- ~ F. g,~,m) k~o

(1.13)

(note that o p e r a t o r (1.12) is a special case o f (1.13)). T h e n the limit of expression (1,13) as t --, 0 exists if and only if x E

D(A m) and gi,m = AiX" In this case the l i m i t is equal to Amx. 2.1.25. [158] A s s u m e that for t > 0

Ctmx:~ -

(--1)'n-kC~kexp(ktA)x .

Then the limit o f e x p r e s s i o n (1.14) as t --* 0 exists if and only if x E D(A m) and is equal to Amx. 2.1.26. [158] Set Tm(t ) := I - - (I - - e x p ( t A ) ) m. Then (i) if x E E and

IITm(t)x

-- x

II = o(tin) as

t ~ 0+, then

Tr~(t)x=x

for all

t~--R+;

(ii) i f x ~ D(Am), then

IIZ~(Ox--xll=O(r") (iii) in the case o f a r e f l e x i v e E (ii) is i n v e r t i b l e .

1050

f o r t~0-+-;

(l.14)

2.1.27. [158] Let A be a generating operator for an OS. Then for fixed m E N we have

A))"x--xii=O;

(i) xEE=~-lim !I(?.R(~,, (ii)

xED(A)=~-Iim I1" Z~II m

(iii)xED(A m

~

A)] ~ x - - x } - - A.x H = 0 ;

'

I

It

~,R(~, A ) x - - k ~ ~ Akx/k ! -- Amx = 0 .

2.1.28. [158] Let x ~ D(Am-1), y E E, and

m--I

Then x 6 D(A m) and Amx = y. If, in this case, y = 0, then k/~(E,

A)x = ~

A~x/~. k for ,~ > max(0,w(A)).

k=o 2.1.29. [158] The following assertions are equivalent (x E E): (i) ~b(t,x) :=

II(exp(tA)

- - l)x II = O(t) as t --, 0; -- o ( ~ - 1 ) with fixed m E N as A --, o~. If E is reflexive, each o f conditions (i) and (ii) is equivalent to the condition x E D(A). (ii)

II[~R(~,A)]~x --x It

2.1.30. [158] For x E D(A m - l ) the following conditions are equivalent:

(i) exp(ta)x- ~] (tA)* x/kt =O(t*/ra!)

as

(ii) ~t~(~, A)x-- ~, Akx/k! =0(~-~) ~.~

~, -->- OO~

II

'

II

/-+0+;

~=0

If E is reflexive, each of condtions (i) and (ii) is equivalent to the condtion x E D(Am). 2.1.31. [158] For a fixed x ~ E the following conditions are equivalent:

(i)

[[(exp(tA)--l)'xll=O(t ~'') as t - + 0 + ;

(ii) I[[;~R(L,

A)--l]"xl[=O(~. -~) as ~--~c~;

IlI

J/li

as

~,--.-~ o o ,

When x E D(A m - l ) each of these assertions is equivalent to the assertions o f 1.30. 2.1.32. [158] If x is such that

li(exp(tA)--exp(sA))x I ! . . < L ( x ~ , then

s, tER+,

~[exp(tsg,)--exp(tA)]xll~L(x)x v2- f o r t > 0. 2.1.33. [158]. Let H be an infinite-dimensional Hilbert space, let P E B(H) be a projector o f finite rank, and let

exp(tA) be such that

(I--P))

supHexp(tA) [I.~-~co and t E ~-+

supllexp(tA)exp(--tP ) I [ ~ o o . Then closure o f set t E~-

O~(exp(tA)" t>o

is a proper e x p ( t A ) - i n v a r i a n t subspace of H. 1051

2.1.34. [92] If two operators A and B are self-adjoint and the operator e x p ( - - B / 2 ) e x p ( - - A ) e x P ( - - B / 2 ) is bounded, then the operator A + B is b o u n d e d below and Ilexp ( - -

(A-q-B)) II~llexp ( - - B / 2 ) exp ( - - A ) exp (--B/2) fl.

Definition 4. A sequence {Xk}keli~ C E is said to be Cesaro summable if the limit

C,--~

~(1--1kl/(n+t))x ~

x~: = l i m

k~Z

n~co k = - - n

exists. 2.1.35. [420] Let equivalent:

Ae~(M, to), xeE, (,oeC

and {~'o + ik 9 k E Z] C p(A). Then the following conditions are

(i) x 6 ~ ( e x p ( 2 ~ o ) l - - e x p (2hA)) ; (ii) the limit

C,--XR(~o+ik, A)x

exists;

k~Z

(iii) the limit

2~t

C,--~R(~o+ik,

A)Q~(;0)x

xdt.

exp(--ikt)exp(--t$)exp(tm)x

exists, where Qk(~): = ( 2 r t ) - '

~Z

~0

Ifthereexistsanr>0suchthat

~--~Cl--Xt~(~+ik, A)x

isanalyticinA(r):={z~C'k--~'ol
kEz the function ~"--* (exp(27r~')I - - exp(27rA))-lx is analytic in A(r). 2.2. O n e - p a r a m e t e r Groups of Operators. Definition 1. A o n e - p a r a m e t e r group of operators, or simply a group, is a family o f operators satisfying the conditions of Definition 1.3.1 with 1~+ replaced by R. 2.2.1. [35] If an OS exp(tA) can be extended to a group (sometimes we say can be imbedded in a group), the extension is unique. We then use the same notation exp(tA) for the group. T H E O R E M 1. [35] For an operator A E L(E) to be a generating operator of a group exp(tA), it is necessary and sufficient for

Ae~(E)

and there exist constants M and co such that {A e C : Re A > w} c p(A) and

II R Q,, A) ~ II "-<

M (I Re ~, ,--o)~

for I Re 7, J > o~, kEN.

2.2.2. If A E B(E), then a simple example of a group of operators i s g i v e n by the formula t~o

exp(tA)=~(tA)~/kl,

t~R.

2.2.3. [288] If, for an OS exp(tA) and every t > 0, the operator $(t) := [exp(tA)] - 1 ~ B(E) exists, then $(t) is an OS with generating operator - - A that can be imbedded in ep(t), t ~ R, in the following manner: ~ e x p ( t A ) for t > O , {])(t) ~-= [ ~ (#)

for

t
We will denote the set o f operators generating a group with bound (1.1) for t > 0 by ~9~(M, r 2.2.4. [288] If 0 Ep(exp(toA)) for some t o > 0, then 0 E p(exp(tA)) for all t > 0 and the OS exp(tA) can be imbedded in a group. 2.2.5. [126] Consider an operator exp(toA ) -- 1 that is completely continuous for some t o > 0. Then the OS exp(tA) is invertible for some t > 0 and it can be imbedded in a group. 1052

2.2.6. [181] Let exp(tA) be a group of operators with unbounded A. Then there exists a number # such that for an), c~,/3 (0 _< c~,/3 < oo) we have

, ~ppl31(ii exp ( t A ) - - / ' I I - I - e , , ) > O.

2.2.7. [ 153] Let exp(tA) and exp(tG) be two groups of operators that commute with each other, and let x be such that ~cfD(A~) NO(P2); A]c, Q~cfiD(A)ND(G); AG.Tc=OA:c. Then IIA~c-iOxli~<2ltxllm(o)/p+21/~X h(:c)ra(1/p)/p+4l[Ax+iOxllm(p).lnp forallp>_ 1, where It(x): = l l A ( A x + i O x ) l ] +llB(Ax+iGx)l], r e ( r ) :

=sup{llexp(tA)exp(sG)ll:t, seR;

lt4-ist<~r f o r r > 0 } . 2.2.8. [153] Under the conditions of 2.7 we have

max(Ii

II, 11021!)~<11~ II ,n(o)lo+l/~h(x),n(110)lo+ +HA~c+iOxit(ll2-}-2ra(p)lnp), p > l .

Ax

2.2.9. [153] Given the conditions of 2.7 and additionally m(p)/p ~ 0 as p ~ oo, then for all L, e > 0 there exists a 6 > 0 such that for I] ~ 11, h(~) < L, the inequality I1A~ § iG~ II < ~ implies IIA~ - iG~ II < ~ if, in addition to the conditions of 2.7, we have m(p) _< m for p > 0, then for p >_ 1 we have

f4rn.I]Axq-iO~i I, ie II~cllq-V'2h(x)~2.11A~-t-iO~Cli; < 14m" ]l A~c -1-iOx I1(! + In [11-~ II+ FStt (x))/(2 II A.~ + iO~ li)]) otherwise.

Definition 2. The set

~(A) : --={xeD(A**) : ~a, c>O: llA'xll~cahn ~, keN} is called the set of analytic vectors of the operator A, and

~c(A) : ={xeD(A**) : V a > 0 a ' c > 0

:

IlA'xll<~cahn,, keN}

is called the set of integral vectors of the operator A. It is clear that for any Ae~'(E) we have ~,(A)~_~(A)~_D(A| 2.2.10. [24] If B E B(E), then ~[(A)~D(A**) -~--~,(A)-----E. 2.2.11. When A is unbounded, the set D(A ~~ may contain only the null element. However, if Ae~(M, o ) , then

D(A| 2.2.12. [86] If AE$'~(M, 0 ) , then ~o(A) is dense in E. 2.2.13. [22] There is an example of an operator AE$'(M, 0), such that We define

D(A, n) : ={xeD(A|

~(A) --{0}.

: ~lc=c(x, n) > 0 , IIAhxll~cn ~, keN}

and

EXP(A) : -----{xeD(A| : ~ta, c>O : llAhxll~ca h, k~N}. The set EXP(A) is called the set of vectors of exponential type. 2.2.14. [22] The set D(A,r/), provided with the norm il x t!o(~,nl: = sup rl-k I] Aax ]] ' is a Banach space. kEN 1053

2.2.15. [22] T h e following propositions are true: (i) D(A,rh) _C D(A,r/2 ) and [[X]iO(A,rl,)~!! )gll,O(.4,rh) for x E D(A,r h) and rh _< rL~; (ii) the set D(A,r/) is invariant under A and 1[ Ax !ID~ ,~ ~< q rl X I:O(~,n~ for x ~ D(A,r/); (iii)

U

(r(AIDIA,~))C_a(A)

and

It P, (~., A)x ho(A,n)--<[l R ( k , A)I} X ilx.!ola,w

for A ~ p(A) and x E D(A.r/);

k=l

(iv) if

EXP ( A ) = E ,

any element x ~ E can be represen.ted in f o r m

x=

~

x~ , where

xkED(A, k),

#=1

2 ilxkllol-~.k)

<

~

, and

il xli--- in~

2 il

XkIIO(A,k),

where inf is taken over all possible representations of x ~ E.

Moreover,

D(A)--

{xEE:x= ~.,x~, k~l

} k=l

and co

A,x--~AiO(A,k)Xk, xED(A).

2.2.16. [22] If Ae$'I~(M, 0) , then E X P ( A ) = E . 2.3. Differentiable Semigroups of Operators. Definition 1. An OS exp(tA) is said to be differentiable beginning with t o (or to become differentiable) if for any x E E the function t ---, exp(tA)x is differentiable on (to,OO). An OS is said to be differentiable if t o = 0. Concerning the properties of differentiable OS's, as well as criteria for whether an OS becomes differentiable, see [60, 341 ]. 2.3.1. [341] If there exist constants C > 0 and 6~ > 0 such that

llexp(tA)--llt~2+C.t.tnt,

0
(3.1)

then the OS exp(tA) becomes differentiable at 3M/C, where the constant M is determined by (1.1). 2.3.2. [419] If an OS exp(tA) becomes differentiable, then the operator exp(tA) is surjective for all t ~ 1t_~. 2.3.3. [419] Assume that AEff(/VI, co) and the operator exp(toA) is not surjective for some t o > 0. Then the OS exp(tA) is not surjective for any t > 0, and in this case the OS exp(tA) does not become differentiable. 2.4. Analytic Semigroups. By E(0) := {z E C : hrg zl < 0} we denote a sector of the complex plane C. Definition 1. An OS exp(tA) is said to be an analytic semigroup with angle 0 ~ (0dr/2), or simply analytic, if it can be extended to an operator function exp(zA) that is analytic in E(0) and strongly continuous at zero on any ray inside E(0). If for any positive ~b < 0 we have ][ exp(zA) [[ < M~ < ~ , z ~ 2(~b), the semigroup is said to be bounded analytic. 2.4.1. An operator function exp(zA) preserves, in P,(0), the semigroup p r o p e r t y

exp( (zl+z2)A)=exp(zlA)exp(z2A),

zl, z2EY(O).

In E(O) with to > w(A) the bound Ilexp (zA)It~M exp (~o .Re z) (] arg zl 401 < 0 <.-t/2).

(4.1)

remains valid. We will denote by 9~(0, o ) the set of generating operators of analytic semigroups with angle 6 that satisfy (4.1).

1054

T H E O R E M 1 ([59, 60]). For an OS exp(tA) the following assertions are equivalent: (i) the OS exp(tA) is analytic; (ii) there exist M and co ~ R such that for Re A >__co > co(A) the resolvent R(A,A) is d e f i n e d and

(4.2)

IIR(x, A)IteM. I~--~ I-L (iii) the OS exp(tA) is d i f f e r e n t i a b l e and for its derivative we have the bound

l[ aTa exp(tA)il--<M't-~'exp(tot),

tER+,

(4.3)

for some M, co ~ R;. (iv) the generating operator of exp(tA) has, for some "~o and e > 0, the bound

IIA(1--ZA)-klI~C/(kM,

(0<;v<M,

keN, k X ~ l ) ;

(v) for an OS exp(tA) the generating operator has, for some M, w, p > 0, the b o u n d

I d)'m+~

2.4.2. [158] For an analytic OS exp(tA) conditions 1~t _. D(A) and II tAexp(tA)l] -< M for t >_0 are equivalent to the statement that OS exp(tA) can be analytically continued to sector s and condition lim~_oexp (~A)x=x(~6 Y ( 0 ) ) is satisfied. Also, when 0 < t _< 1 and a E (0,1/(me)) we have

A exp (tA) = 2"~7"~t

f

exp(:A)/(;--t)2d'-"

~ , - t I=~' sin

2.4.3. [158] If an analytic OS exp(tA) has the bound [{exp (;A) 'l -,< M ~

for

e / larg~.i--<-7~f,

;s162

where bl is given by 4.2, then for the ~ given in 4.2 we have [IA exp (tA)ll<~M,/(t-sin ~). 2.4.4. [44] For an analytic OS exp(tA) we have (i)

il

II

(exp(tA)) =!iA~exp(tA)~l..<Mke~t/t ~, t . > 0 ;

(i i) i' Ak(exp (tA) - - exp (sA))il 4 M~ (t~ - - s ~) e ~'<,' [k (ts)q, 0 < s < t

where M k = M k . k k and w > co(A). 2.4.5. [193] A generating operator A generates a bounded analytic OS exp(tA) with angle 0 if and only if Z(0 + _.) z p(A) and for any ~ E (0.0) there exists a constant M~ such that ;i R(),,A) i[ -< M~,, I~l for I E E(~b + rr/2). 2.4.6. [44] An OS exp(tA) is analy'tic if and only if there exist an % > w(A) and an bl >_ 0 with the property

IiR(m,+in, A)ll<~MIInl

fo~ a n

tieR.

1055

2.4.7. [293] If A generates a bounded group of operators exp(tA), then - - A 2 generates a bounded analytic OS with angle r / 2 . T h e statement remains true if the word "bounded" is omitted. 2.4.8. Assume we are given an OS exp(tA) that is analytic in ~(0). Then the o p e r a t o r function U(t) = exp(t exp(i~o)A) is an OS f o r all

U(t>=~zi I e - ' A ( z A + t I ) - ' d z , F

where 1" = 8[{x ~ C : Izl = r > 0} u ~(0)]. 2.4.12. [292] If E is reflexive, then in 4.11 condition of denseness m a y be replaced by condition 2.4.13. [162] T h e following statements are equivalent: (i) the OS exp(tA) is analytic; (ii) there exists a n u m b e r ~" ~ C, ~'1 = 1, such that

lira in~ {[I ; x - - exp (tA) x , ~,,x ~-~c,~ tt x

t-+O+ x

II =

.]P(A) = { 0 } .

1 } > o;

(iii)

li oi sup {t II A exp (tA) X !I:xED (A), II x [t = 1 } < oo ;

t-+O+

(iv) the o p e r a t o r

(A OA) generates an OS on E x E;

(v) there exist a constant C and a n u m b e r b > 0 such that

nil (~.R (~,, A)"-- (~.R (~,, A))"-'II~C

for A > nb, n

N; (vi) there exists a polynomial q such that

lim [J q(exp(tA))ll < sup {I q ( z ) l : z e C , t-*0+

l,zr----l};

(vii) the strict inequality

< L(q):=sup{[q(z)l:z6C, I z l = l } is satisfied for all c o m p a c t ,3~ER and all polynomials q(z) such that Iq(l)l < L(q); (viii) for any (" ~ C such that ~'1 = 1 and ~" #= 1, for sufficiently small t the o p e r a t o r fI - - exp(tA) is invertible. 2.4.14. [162] If we have, for an OS exp(tA), the inequality

lira Iiexp (~A)-- Z II <2,

t~0+

1056

then the OS exp(tA) is analytic. T h e converse is generally not true, but does hold, for e x a m p l e , in the case of a uniformly convex E or a contraction OS. 2.4.15. [59] If an OS exp(tA) is analytic with angle 0, then t~c, C_ D(A ~ ) for all c o m p l e x a E ~(0). 2.4.16. [59] Let exp(tA) be a b o u n d e d analytic OS and let 6 = 7r/2 + ~b, with ~b as in 4.5. T h e n

e x p ( t A ) = ~ T1

I eZ~/?(~"

A)dX,

(4.4)

I'tp

where F ~ - ~ F , ~ U I ' ~ _ ,

r,o+=pe +-~ ( z / 2 < q 0 < 6 ,

0
~).

2.4.17. [158] Assume we are given an analytic OS exp(tA). T h e n (i) ~b(t,x) = O(t '~) for t ---, 0+ and 0 < a < 1 if and only if II A e x p ( t a ) I I = o ( t ~ - ~ ) for t ~ 0+, which, in turn, occurs if and only if II A~exp(tA) I! = o ( t ~-2) for t ---* 0+; (ii) if ~b(t,x) = O(t), then IAexp(tA)l = O(log(l/t)) and [I A2exp(tA) II = o(t for t ---, 0+. 2.4.18. [28] If A generates a b o u n d e d analytic OS, the set of integral vectors ~o(A) is dense in E. 2.4.19. [281 Assume that A generates an analytic contraction OS exp(tA). Then go(A)= fl ~ ( e x p (tA)), there t>0

exist inverse operators for the OS exp(tA) that are defined on ~c(A). , and

[exp(tA)] -I

x = Z tk(--A)~x/le!

for all

k=0

x~aAA) 2.4.20. [169] Let A E ~ ( r t / 2 , 0 ) In this case EXP(B) = EXP(A).

and B 2 = A. Then for any r/> 0 the Banach spaces D(B,r/) and D(A,r/2) coincide.

2.5. F r e d h o l m Semigroups of Operators. G i v e n an operator Ao~(E), we denote the dimension of the kernel ,h~ by n u l ( A ) : = d i m A ~ . Defect of an operator A is codimension of ~(A), i.e., d e f ( A ) = d i m ( E / N ( A ) ) . If nul(A) or def(A) is finite, then the index of the operator can be defined: ind(A) := nul(A) - - def(A). Definition 1. An operator A ~ ( E ) is said to be F r e d h o l m (Noetherian) if l~(A) is closed and nul(A) and def(A) are finite. If, h o w e v e r , only one of the numbers nul(A) or def(A) is finite, the o p e r a t o r is said to be semiFredholm. Definition 2. An OS exp(tA) is said to be s e m i - F r e d h o l m if the operator exp(tA) is s e m i - F r e d h o l m for all t >_ . 2.5.1. [219] Assume t o > 0. T h e following statements are true: (i) if nul(exp(toA)) < ~ , then nul(exp(tA)) = 0 for any t >_ 0; (ii) if def(exp(toA)) < e~, then def(exp(tA)) = 0 for any t >_ 0; (iii) if an operator exp(toA ) is Fredholm, then the operators exp(tA) are bijective for all t E R+; (iv) if the o p e r a t o r exp(toA ) is s e m i - F r e d h o l m , then for all t > 0 the operators exp(tA) are s e m i - F r e d h o l m and nul(exp(tA)) = nul(exp(toA)) and def(exp(tA)) = def(exp(toA)). T H E O R E M 1. [219] Let E be a reflexive space. T h e n an OS exp(tA) is s e m i - F r e d h o l m if and only if for all # > co(A) and 1 _< p < oo

inf ~ e-~"ilexp(tA)xjlPdt+ inf I e-"':lexp(tA)*Y*ll~at>O, Slxll=~ ~ Iiv.~t=ldJ

where exp(tA)* is the operator adjoint to exp(tA). 2.5.2. [219] A differentiable OS with an unbounded generator cannot be s e m i - F r e d h o l m . 2.5.3. [219] If an OS exp(tA) is s e m i - F r e d h o l m and has an u n b o u n d e d generator, then li.'-m!1e x p ( t A )

-- I 1!>~2.

t~0+

where

We define reduced minimal modulus of an operator A ~ L(E) as 7(A):=inf{llAxH/dist(x, dist(x,./F(A)) is the distance f r o m the element x to the kernel ./r(A). 2.5.4. [44] The range of an operator AC~' (E) is closed if and only if 7(A) > 0.

.IF(A)) : x~AP(A)},

1057

2.5.5. [219] If an OS exp(tA) is s e m i - F r e d h o l m , then (i)

sup t -~ In [7 (exp (tA))] = lim t -~ In [7 (exp (tA))] ~< o~(A); f>O

t~

(ii) there exist constants 6 > 0 and a such that 6- exp (~. t) ~ 0 (and consequently for all t >_0 - - s e e 5.1). Then, in the h a l f - p l a n e SF(A):---- {zEC : R e z < sup t -~ In [~f(exp (t A))I} />0

we have (i) AI - - A is s e m i - F r e d h o l m for all A E SF(A); (ii) nul()d - - A) and def(XI - - A) are constant for )~ ~ SF(A), where nul (XI--A) nul(XI--A) def (XI--A) def (~.I--A) (iii) ~rt~

I'I(XI--A)--'I(~,tI--A)1~17.--~,11

= O, i f nul(exp (toA)) < o o , >0, =0, >0,

if if if

nul(exp (toA)) = o o , def(exp (toA))
for,X, ,X1 ~ SF(A).

2.5.7. [219] If an OS exp(tA) is s e m i - F r e d h o l m and the resolvent R(A,A) is c o m p a c t , then exp(tA) is bijective 0. 2.5.8. [219] The following statements are equivalent: (i) ][ e x p ( t A ) x II >- exp(pt)II x I1 for all x ~ E and t >_ 0; (ii) 11(AI - - A)x II ->

_ p. 2.5.9. [219] For AEI~(I,0) the following statements are equivalent: (i) the o p e r a t o r exp(tA) is isometric f o r e v e r y t > 0; (ii) II (11 - - A)x II -> - I II x I1 f o r all X _< 0 and x ~ D(A); (iii) f o r e v e r y x ~ E there exists an x* ~ 8(x) such that Re(Ax,x*) = 0. When E = H the assertions of 5.8 and 5.9 m a y be restated in terms of the numerical range of the operator

W(A) : = {(Ax, x) : Ilxll = 1,

xeD(A)}.

2.5.10. [219] T h e following statements are equivalent: (i) the OS exp(tA) can be i m b e d d e d in a group and -~(exp(tA)) >_ ePt; (ii) Re W(A) > p; (iii) II (1i - A)x II -> (P --I)II x 11 for all ,X < p and x ~ D(A). When these statements are true we have

lira ~ (exp(tA))= 1. t~0

2.6. Spectral Properties of Operator Semigroups. G i v e n an operator A6.~(E) we will, as ususal, separate its spectrum into three sections, the point s p e c t r u m Pox(A), the continuous s p e c t r u m Ca(A), and the residual Rot(A). Definition 1. By the F r e d h o l m domain of an operator AE~(A) we mean the set f~F(A) of all points z ~ C for which the o p e r a t o r zI - - A is Fredholm. Definition 2. T h e c o m p l e m e n t of flF(A) is called the essential s p e c t r u m of the operator A 6 ~ (E) and is denoted by Eg(A). 1058

2.6.1. [178] It is clear that Ea(A) _ a(A) for all

A6W (E).

Definition 3. We denote be EFe(A) the set of all z ~ C for which the o p e r a t o r zl - - A is not s e m i - F r e d h o l m . Definition 4. The set of all points ), ~ C for which the operator AI - - A is not injective or ~(~,I--A) is not closed in E is called the a p p r o x i m a t i v e s p e c t r u m of the operator A~'(E) and denoted by Aa(A). 2.6.2. [126] For any AE~'(E) we have

Po(A)~_Ao(A) and o(A)=Ao(A)URo(A). 2.6.3. [126] For any

A6CG(E)

we have

(i) the b o u n d a r y Oa(A) of the set a(A) is contained in Aa(A); (ii) Rot(A) = Pa(A*), where A* is the adjoint of A. As usual, for an operator B ~ B(E) we define the spectral radius r(B):=sup{l~l

: Z,6o(B)},

and the essential spectral radius

Er(B) : =sup{l~,l :ZEEo(B)}. Definition 5. T h e measure of noncompactness of a K u r a t o w s k i set fl c E is the quantity cz(fl) := inf{d > 0: there exists a finite n u m b e r of subsets

tO

~k ,--- ~" i--j/j=l'---*-, such that

k

9.C U~ i ,

sup{[[x-Yil:

x , YE.Q,}--.
k} 9 The

]=1

measure of noncompactness of the e m p t y set is taken to be zero. For detailed i n f o r m a t i o n on the measure of noncompactness see, for example, [2]. Definition 6. Let B E B(E). The quantity [I IBI I I=: = i n f { d > 0 is called the measure of n o n c o m p a c m e s s of the operator B. 2.6.4.[126] For any B ~ B(E) we have

Er (B)= im III Itl = im II

: ~z{Bfa)~,d.cz(Q)

for any bounded f2 E E

Ii','*, (6.1)

where tt B II* = inf{ I1B - - K It " K ~ Bo(E)}. 2.6.5. [126] Let A E , ~ ( M , ~ ) . Then

S(A)<~ (A).

(6.2)

2.6.6. [126, 424] Inequality (6.2) may be strict. 2.6.7. [126, 401] The equation co(A) = S(A) holds in each of the following cases: (i) the OS exp(tA) becomes compact; (ii) the OS exp(tA) becomes differentiable; (iii) the OS exp(tA) is analytic. 2.6.8. [126, 178] For A6,~(M, co) we have

r(exp(lA) )~---exp(oJ(A)t), t>~O. Definition 7. T h e essential growth index of an OS exp(tA) is the quantity

E ~ ( A ) : = inf ~-1 In !I]exp(tA)ilt ~ = l i m t - ' . I n

jltexp(tA)lil ~.

1059

2.6.9. [178] Let AO~(M, o ) . Then E~0(A) _
Er (exp (tA)) = exp (Eo (A) t) npx Bcex t>~0. 2.6.11. [178] We set SI(A) :- sup{Re A : A E e(A)\Ea(A)}. Then (i) sup{ReX :X6Eo(A)}<~.Eo(A)); (ii) o(A)=max{S(A), Ear(A)}--max{St(A),

Eo(A)}.

2.6.12. [101] Let Aeff(M, co) . If infinity is an isolated sigular point of the resolvent R(A,A), then it is an essential singularity. If A E B(E), then we have exp (ta (A)) --a (exp (tA)) \ {0},

(6.3)

which is called the spectral mapping theorem for operator semigroups. Here exp(tfl) := {A ~ C : z ~ f2 : A = exp(tz)}. 2.6.13. [126] For an a S exp(tA) Eq. (6.3) holds if, for example, (i) the a s exp(tA) is compact; (ii) the a S exp(tA) becomes differentiable; (iii) the a S exp(tA) is analytic; (iv) the a S exp(tA) becomes uniformly continuous. 2.6.14. [1261 I f I[exp(tA)II -< c . t k for a group exp(tA) for some C > 0, k ~ N, and all t ~ R, then we have the following relationship, which is weaker than (6.3):

a(exp(tA))=exp(ta(A))

for a l l

tER.

2.6.15. [126] For an a S exp(tA) we have the inclusion

exp (to (A)) ___a(exp (tA)) ear t~>0. In more detail, (i) exp (tPe (A) ) = Po (exp (tA) ) "\ (0}; (ii) exp (tRa (A) ) = Re (exp (tA) ) \ {0}; (iii) exp (tAa (A) ) ~ A e (exp (tA) ); (iv) exp (tEa(A)) ~ E a ( e x p (tA)); (v) exp (tEFa (A) ) ~_EFo (exp (tA) ).

2.6.16. [126] Let AE$(M, ~o). Then for all t E R+ (i) if A E Co(A) and none of the numbers Ak = ), + 2rik/t, k E Z, belongs to Pa(A) u RO(A), then exp(tA) E Co(exp(tA)); (ii) if A ~ RO(A) and none of the numbers Ak, k ~ Z, belongs to PO(A), the exp(t),) ~ Ra(exp(tA)); (iii) if exp(tA) ~ Ro(exp(tA)), then none of the Ak, k E Z, belongs to Pa(A) and there exists a k ~ N such that ~ERa(A) ; (iv) if A 6 PO(A), then/~ := exp(tA) ~ o(exp(tA)). If, however, # E Pa(exp(tA)), then thereexists a k ~ Z such that h/-6Pa(A) and ,0 t t / ~ J - - e x p ( t A ) ) t ) = V .,f((kk l - - A ) 1) for all integers l -> 1. Here ~,/ .o.u is the smallest closed subspace containing all of the subsets if2k. 2.6.17. [ 178] Assertion (6.16) (i) cannot be reversed, or, more accurately, it is possible that exp(tA) 6 Co(exp(tA)) at the same time as A + 2 r i k / t E p(A), k 6 Z.

1060

2.6.18. [126] Let E = H be a Hilbert space and {exp(tA): either

Onik

~ k = ;~ + @

fi~(A)

AE~'(M, co). Then for each t e 1/+ the set o(exp(tA))\{0} =

for some k E Z or the sequence II R(Ak,A)11 keZ is unbounded}.

2.6.19. [126, 178] For an OS exp(tA) we have (i) ~ ( e x p ( t A ) ) = o ( e x p (tA)| *) for all t ___0; (ii) R~(exp(tA))=Pg(exp(tA)~)=Po(exp(tA)*)~Rg(exp(tA))

UPa(exp(tA))

for all t e 1~+ ;

(iii) o ( A ) = o ( A ~ ) = o (A*); (iv) R~(A)=PJ(A~)=P~{A*)cR~(A)UPcr(A); (v) S(A) = S(A*) and w(A) = co(A*).

2.7. Stable Operator Semigroups. 2.7.1. [101] There is a w e l l - k n o w n classical example in which a(A) = ~ and IIexp(tA)I[ = e x p ( r t / 2 ) , t e I~+. 2.7.2. [424] When E = H is a Hilbert space, for any real a < b there exists an OS exp(tA) such that S(A) = a and I[exp(tA)Ii = exp(bt), t >_ O. These examples indicate that there is a d i f f e r e n c e between the finite and infinite dimensional cases when the operator A is represented as a matrix. We set co(x, A ) : = i n f {co6R :

Ilexp(tA)xll~M

exp(cot)

for some

M

and t~>0} is the exponential growth bound for the function exp(tA)x; c o l ( A ) : = s u p {~o(x,

A) : xED (A ) }

is the exponential growth bound for the solution of Cauchy p r o b l e m (1.2.1) with n -- 1. It is clear that w(A) = sup{w(x,A) 9 x E E}. 2.7.3. [126] For any x ~ E and any OS exp(tA) we have co(x, A ) = lira t -I In ]lexp

o~(x, A ) = i n f

Re),:

(tA)x

il;

exp(--'~.t)tiexp(tA)xHdt exp(--Xt)exp(tA)xdt

co1(A)=inf R e k :

exists

e x i s t s as

0

an improper integral for any

=inf

e-Z'i:,exp(tA)x['dt

Re),:

X~E}~

e x i s t s for any

xeD(A)}"

0

If A ~

[O), then, for x ~ D(A) we have co (x, A ) = inf {Re ~,: i e - u exp

(tA) Axdt

exists as an i m p r o p e r integral}.

0

2.7.4. [126] It is clear that S(A) _< wl(A ) _< co(A), and there are examples in which both inequalities are simultaneously strict. Definition 1. An Os exp(tA) is said to be a) stable if [1exp(tA)x I[ ~ 0 as t --. oo for all x ~ D(A); b) u n i f o r m l y stable if a) is satisfied for all x E E; c) exponentially stable if col(A ) < 0; d) u n i f o r m l y exponentially stable if w(A) < 0. 2.7.5. [126] Let A ~ B(E) and S(A) < 0. Then an OS exp(tB) is u n i f o r m l y exponentially stable. In the case of an unbounded generating operator this is generally false. 1061

2.7.6. [ 126] T h e following statements are equivalent: (i) the OS exp(tA) is stable; e~

(ii) the subspace

A~

and the integral

~exp(tA)xdt

exist for all x E D(A).

0

2.7.7. [126]. T h e following assertions are equivalent: (i) the OS exp(tA) is stable and AE~?(M, 0) ; (ii) the OS exp(tA) is u n i f o r m l y stable; oz

(iii) AE~(M, 0). and there exists a subset I) c E that is dense in E and such that integral

l exp (tA)xdt

exists

0

for all x E I). We write b(x):=inf

Re~,:

e-Z'exp(tA)xdt

exists

2.7.8. [332]. G i v e n an OS exp(tA) and x ~ D(A), we have (i) w(x,A) = max(b(x),b(Ax)); (ii) if b(Ax) __. 0, then 0 ~ o ( x , A)~b(Ax) = i n [ {oER: Ilexp(tA)x--xll~Me" for some M and all t ~ R+). (iii) if , d e ( A ) = { O } , then ~o(x,A) = b(Ax); (iv) if b(Ax) < 0 then b(Ax)= i n [ {oeR : Ilexp(tA)x--vll ~<Me -' for some M and t e 1~+), where Moreover, if -4~ = { O } , , then to(x,A) < 0.

yeA~

Definition 2. For x e E we introduce the following notation: 7(x,A) := inf{a E R: the f u n c t i o n ), ---, R(A,A)x has a holomorphic continuation to the h a l f - p l a n e {A e C 9 Re A _ a}], 7o(X,A) := inf{a e R : a > ~/(x,A) and the function R(A,A)x is bounded in the h a l f - p l a n e {A ~ C : Re A _> a)}; 71(x,A) := inf{a 6 R : a > 7(x,A) and the function R(A,A)x u n i f o r m l y converges to zero with respect to Re A as Ilm AI---, oo t h r o u g h o u t the h a l f - p l a n e {A ~ C : Re A >_ a + e}, e > 0}. 2.7.9. [332] For any x E E we have "/(x,A) _< %(x,A) _< 71(x,A) _< w(A). If, h o w e v e r , x e D(A2), then 7l(x,A) _< ~o(x,A) _< max{7(x,A),70(A2x,A)}. 2.7.10. [332] Assume that we are given a positive u n i f o r m l y bounded OS exp(tA) in a space E with a normal reproducing cone. I f lim R(~,, A)x exists for all x e D(A), then the OS is u n i f o r m l y stable. ~,~0 +

2.7.11. [126] An OS exp(tB) is u n i f o r m l y exponentially stable if and only if S(A) < 0 and there exists a t o such that Aa(exp(toA)) is inside the unit circle. 2.7.12. [126] T h e following statements are equivalent: (i) ~o(A) < 0; (ii)

ltmllexp(tm)tl=0

;

t -.*-oo

(iii) f o r some t o > 0 we have II exp(toA)II < l, oo

(iv) for any (some) p > 1 the integral

~llexp(tA)xtlvdt 0

2.7.13. [126] G i v e n an OS exp(tA) that is stable. Then (i) S(A) _< 0 and Re A < 0 for any A e Pa(A) u Ra(A); (ii) the limit lira k R ( h , A)x exists for any x ~ D(A). h~0+

1062

exits for an), x ~ E.

2.7.14. [126] Assume that exp(tA) is a u n i f o r m l y b o u n d e d analytic OS. T h e n the following statements are equivalent: (i) 0 $ Pe(A) u Ra(A); (ii) the OS exp(tA) is u n i f o r m l y stable. 2.7.1S. [377] Let S(A) _< --6 < 0. T h e n (i) if IIR(A,A) I[ = O(~r as IIM,~[-. oo, Re)~ > - - 6 , and k > - - 2 , then there exist 0 < # < 6 and g(#) > 0 such that [[A-r for any x ~ E, t _> O; (ii) if (i) is satisfied for k _> --1, then there exists a # E (0,6) such that

fl e"t ( A-(k+l) exp(tA)x,

X*

) 12dt-.< K(~, x , x*)

0

for all x E E, x* E E*. 2.7.16. [126] Assume we are given a positive OS exp(tA) and D ( A ) _ := --D(A)+. T h e n wl(A ) < 0 if and only if E+ c AD(A)_. 2.7.17. [126] Assume we are given an OS exp(tA) that is positive and bounded. If a subset /~-e'-E+ that is total in E is such that lim R(~., A)x for all x E ~ - , then the OS exp(tA) is u n i f o r m l y stable. ~0+

2.7.18. [377] Assume that S(A) _<--6 < 0 and exp(tA) becomes differentiable. T h e n there exist constants # ~ (--8,0) and K --- K(#) such that ]l (exp(tA)I1 -< K e x p ( - - # t ) for t >_ 0. 2.7.19. [124] Assume that E is reflexive, AE,~ (M, 0), P a ( A ) N i R = ~ , and the set o(A) n iR is countable. Then the OS exp(tA) is u n i f o r m l y stable. 2.7.20. [126] There are examples in which an OS exp(tA) is u n i f o r m l y stable, Pc(A) n ill = ~, but e(A) n iR 2.7.21. [124] There is an example of an OS that is not u n i f o r m l y stable in a reflexive E but is such that Pc(A) = and o(A) C_ JR. 2.7.22. [124] Assume AE~(M, 0), Ro(A)NiR=fg , and the set o(A) n iR is countable. Then the OS exp(tA) is u n i f o r m l y stable. 2.7.23. [124] 7.21 is precise in the following sense: (i) there exists a unitary group exp(tA) such that e(A) _C iR, Re(A) -- ~ and at the same time this group is unstable; more precisely, the condition exp(tA)x --. O as t --, oo implies x = O; (ii) there exists an OS exp(tA) such that c~(A) = 0 and Re A < 0 for A E e(A), but the exp(tA) is not u n i f o r m l y stable; (iii) let E = [C,oo) and (exp(tA)f)(x) = f(t + x). Then or(A) = {A E C : Re A _< 0} and the OS exp(tA) is u n i f o r m l y stable. 2.7.24. [124] Assume that E is reflexive, AG~ (M, 0), and the set e(A) n iR is countable. Then E is the direct sum of invariant subspaces Est and E~p, where Esr : -----{xEE : !ira exp (tA) x = O}, E ap: = span {x6D (A): A x = ),x )~E t~vc

iR.}.

M o r e o v e r , the restriction of the OS exp(tA) to Eap can be i m b e d d e d in a group on Eap. 2.7.25. [124] G i v e n an OS exp(tA) that becomes continuous with respect to norm, if Re A < 0 for all ,~ E a(A), then the OS is u n i f o r m l y stable. 2.7.26. [126] Assume that a positive OS exp(tA) is u n i f o r m l y bounded and becomes continuous with respect to norm. Then the following conditions are equivalent: (i) the OS exp(tA) is u n i f o r m l y stable; (ii) 0 ~ Ra(A), i.e., A~ *) = { 0 } ; . When E is reflexive, conditions (i) and (ii) are equivalent to (iii) 0 ~ Pa(A), i.e., al'~(A~={O; . 2.7.27. [124] If, in 7.22, condition A E ~ ( M , 0) is replaced by assumptions S(A) _< 0 and sup ii exp ( t A ) B i] < t-.70 . where B ~ B(E) is an operator that c o m m u t e s with OS exp(tA), then the assertion is replaced by lira ~,Ie x p ( t A ) .

Bx/--O

for a n y x E E .

1063

2.7.28. [124] Let Re 2 < 0 for all ~ ~ or(A) and sup Ji exp (tA) Ax !i < ~

for some x ~ D(A). Then the OS exp(tA)

t>~0

is stable. 2.7.29. [311] Assume we are given an OS exp(tA) that is uniformly bounded, the set e(A) n iR is countable, and the adjoint A* is such that Pa(A*) n iR = ~. Then the OS exp(tA) is uniformly stable. 2.7.30. [311] Let exp(tA) be a uniformly bounded group and a(A) n iR = ~. Then it is uniformly stable. Let A ~ ( E ) , mENU{0}. For a ~ p(A) we set

tom(A): = i n f {coER:sup iJe-'" exp (tA)R(cz, A)" !l < c~ } t~0

and F,=(A) : = i n f { s > o ( A ) :llR(a-+-ib, A)II=O(lb for b ~ co and a~s}.

1")

2.7.31. [378, 407] Given an OS exp(tA) in a complex Banach space E. Then for any m ~ N u {0} (see 7.15) (i) Wm+2(A) _<Em(A); (ii) if E = H is a Hilbert space, then corn(A) _< -=m(A); (iii) the relation O)m+l(A ) < -~m(A) is satisfied for a broad class of spaces E including, in particular, Lebesgue, Besov, and Sobolev spaces. 2.7.32. [378] There are examples in which (i) OJm+l(A) < com(A) and S(A) < Em+I(A ) < ~m(A), m E N u {0}. (ii) E is uniformly convex and wa(A) < Eo(A) < Coo(A) Definition 3. An OS exp(tA) in a Hilbert space H is said to be weakly LP stable if for any x, y ~ H we have etz

i(exp(tA) x, V)L"dt < o~. 0

2.7.33. [418] Weak L p stability of an OS for some p E [I,oo) implies weak stability for the OS for any p ~ [I pc). 2.7.34. [418] An OS exp(tA) defined in H is weakly L p stable for some p E [1 ,oo) if and only if it is uniformly exponentially stable. 2.7.35. [418] If there exists a p E [1,oo) such that for all x E E

i

llexp(tA)xil" dt < ~ ,

(7.t)

0

then the OS exp(tA) is uniformly exponentially stable. 2.7.36. [418] An OS exp(tA) defined in H is uniformly exponentially stable if and only if for some p E [1,oo) inequality (7.1) is satisfied. 2.7.37. [418] In general, weak LP stability does not imply uniform exponential stability. 2.7.38. [126] Given an OS exp(tA) that is positive. Then, if either (i) the operator (AI -- A) is invertible for some A > 0, or (ii) the operator A is invertible and A -1 _
( i I( e x p ( t A ) x , y * ) I o d t ) ~ t ~

1064

V* ][

(7.2)

with some constant M 0 >__0 if the integral on the left exists. 2.7.40. [418] G i v e n the conditions and notation of 7.38 with p - 1 + q-X = 1, then for all ~o(t) E Lq[0,oo) and x E E the limit T

(qo: =lira T--*-co

exists and

I ,(t)exp(tAJxdt 0

lit(@)[[~M0llxll'l[K0ilLqt0,o~ 1- 9 This statement holds when the space E is weakly sequentially complete (in

particular, when it does not contain Co). 2.7.41. [418] Assume that (7.2) is satisfied and 1 < p _< oo. Then, for any z E C with Re z > 0 we have z E p(A) and for any x E E we have

T

R(z, A ) x - - l i m I e-zt exp(tA)xdt. 'lr-.~ oo 0

2.7.42. [418] U n d e r the conditions of 7.40 and 7.39 we have

IlR(z, A)[["<~M~

for p > 1, for p = 1.

2.7.43. [418] U n d e r the conditions o f 7,40 and 7.39, but with p < oo, we have

S(A)~--B-~---1/(pMo) p For z ~ C with --6 < Re z _< --6

+

Mo-P we have

HR(z, A)II~(Re z+~)-L Definition 4. We define H ~176 := (f(z) 9 f(z) is bounded and analytic when Re z > 0 and has values in E}. 2.7.44. [418] Let E = H be a Hilbert space and R(z,A) E H ~176Then the OS exp(tA) is u n i f o r m l y exponentially stable. 2.7.45. [418] U n d e r the conditions of 7.33 we have w(A) _<--1 / (pM) p with constant V) :Pdt:il xI], IIV H~
Mp: = s u p

{il'

exp(tA)x,

2.7.46. [126] Let the OS exp(tA) be positive, u n i f o r m l y bounded, quasicompact, such that S(A) = 0. Then there exist a positive projector P of finite rank and constants 6 > 0 and M _> 0 such that t>_0.

I!e x p ( t A ) ,

P 1[ _< M exp(--6t) for all

2.7.47. [126] Assume that the OS exp(tA) is positive, uniformly bounded, and continuous with respect to norm; also, assume that the space E is reflexive. Then the limit /5x----- lira e.xp ( t A ) x exists for any x ~ E, where 15 is the positive projector onto

di~

2.7.48. [126] Assume that under the conditions of 7.46 we also assume that p is the order of the pole Ao = S(A) of the resolvent R(A,A). Let P()'o) be the projector onto A~ p) along Yt((~.oI--A)P). Then, (i) for any r/E (0,e) there exists a constant M(r/) such that [[exp(tA)(I -- P(Ao)) [] _<M(r/)exp((~ o - - r/)t) for t E 1~+; (ii) if p = 1, then for any 77 ~ (0,s) we have

Jexp ( -- LJ )exp ( t A )-- P ( k,3 ii --<M ( q )exp (-- ~lt), left..: 1065

(iii) if the OS exp(tA) is irreducible and Ax 0 = AoXo, A ' x 0- = ~oXo , (Xo*,X0) = 1, then for an5, r/E (0,e) and x E

Ilexp(--Xotjexp(tA)x-- <x ,

x o >x 0 I ! ~ < M ( q ) . e x p ( - - v l t ) ,

tER~.

f

Definition 5. An OS exp(tA) is said to be ergodic if

t-I .i exp(sA)xds

strongly converges as t --, co for any

0

xEE. 2.7.49. [278] An OS exp(tA) is ergodic if and only if

E={xEE:exp(tA)x==x,

E,R+}O

xEE:t -I exp(sA)• 0

xd s -+O

as t ---, oo}.

2.7.50. [278] If an OS exp(tA) is ergodic, the following conditions are equivalent: (i) y ~ ( A ) ; (ii) the limit lim R('~., A)y exists; ~.--0+

(iii) the limit

~tmoC1 1

I exp(sA)ydsdfl

exists;

0 0

(iv) for some sequence {a n} that becomes infinite, (iii) holds in the sense of a w-lira. 2.7.51. [278] Assume that an OS exp(tA) is such that t -111 exp(tA)I! -" 0 as t ~ oo and

suo'-'11 sA'X, exp o o

for all x E E and some M < o o . Then each of conditions ( i ) - ( i v ) o f 7.50 is equivalent to the condition

11 y~l( A )~iAiOO(A)). 2.7.52. [278] Let

AES'(M, 0 ) . T h e n the condition . y*E~(A*)

is equivalent to condition

exp (tA)*

sup t>O

9

y*dsll < oo. 2.8. Positive Semigroups of Operators. Definition 1. A subset E+ of a Banach space E is said to be a cone if (i) E+ is convex; (ii) together with any x E E+ the cone E~ contains the ray (tx: t _> 0); (iii) E§ contains no opposing elements; (iv) E+ is closed. Definition 2. For two elements x, y E E, the relation x~y is equivalent to the condition y - - x E E+. Definition 3. A cone E+ is said to be a reproducing cone if E = E+ - - E§ In this case there exists a n u m b e r C > 0suchthatforeveryxEEwehavex=x z-x2,where lixjll -
1066

Definition 7. Let E be a Banach structure, x ~ E. By the modulus o f an element x we mean Ixl := x § + x - , where x + = sup(x,O) is the positive part of the element x, and x - -- sup(--x,O) is the negative part o f x.

Definition 8. A set S c E is said to be order b o u n d e d if there exists an x such that y~x f o r all y e S. Definition 9. A partially ordered space is said to be order complete if any order b o u n d e d set in it has a sup. It is said to be a - o r d e r complete if any countable order b o u n d e d set has a sup. Definition 10. T w o elements x and y o f a Banach structure E are said to be disjunctive if inf{kl, k,[} -- 0. Definition 11. Let A, B ~ B(E). We will say that A<~B, if Ax~.Bx for all x ~ E+.

Definition 12. If 0 _< A we will say that A is positive. Definition 13. A n OS exp(tA) defined in a partially ordered space E is said to be positive if

exp(tA)E.~cE+

f o r any

2.8.1. [126] Assume that an OS exp(tA) is positive and

exp(~.A)x~exp(tA)y

x
t~O. Then

any

t~0.

2.8.2. [178] Positivity of an OS exp(tA) is equivalent to satisfaction o f any of the following conditions: (i) the resolvent is positive for all A > w(A), i.e., R(),,A)E+ c E.~; (ii) the relation

x .~g

implies R(A,A)x _< R(A,A)y for Re ~ > w(A).

Definition 14. A functional y* ~ E* is said to be positive if <x,y*) _>0 for all x E E+. We denote the set of positive functionals on E by E+'. If the cone E+ is normal, then E+" is a cone in E ". 2.8.3. [125, 146] If an OS exp(tA) is positive, then D(A)+ := D(A) f E+ is dense in E+ and D(A*)+*w* is dense in E~'. Definition 15. The spectral bound of an operator A is defined to be

S(A):=sup{Re~.:lEo(A)},

where

S(A)=--oo,

if

o(A)=~.

2.8.4. [244] If an OS exp(tA) is positive, the cone E+ is normal and reproducing. Then, for S(A) < Re ~z _< w(A) Eq. (1.2) holds, i.e.,

R (u, A)x= i e-Utexp(tA)xdt' 0

where the integral exists as a nonsingular R i e m a n n integral. 2.8.5. [178] Given an OS exp(tA) that is positive and a Banach structure E, then

R(p,A)x~R(Retx, A)!x I

for a n

xEE

and R e p . > S ( A ) .

D e f i n i t i o n l 6 . A linear subset J c E is an ideal if it follows f r o m x ~s E, y ~ J , and I x l < l Y [ t h a t x E J . Definition 17. A positive semigroup exp(tA) is said to be irreducible if {O} and E are the only closed ideals that are invariant under the family exp(tA) with t _> 0. Definition 18. If x ~ O , the smallest ideal containing x is called the principal ideal Jx corresponding to x. Definition 19. If Jx is dense in E, then x is said to be quasi-interior. Definition 20. An OS exp(tA) is said to be stricly irreducible if for any quasi-interior for the cone E+.

x-~O. t > 0

, all elements exp(tA)x are

2.8.6. [244] If E is a Banach structure, the following statements are equivalent: (i) exp(tA) is irreducible: (ii) R(),,A) is strictly irreducible for all A > S(A): (iii) RO,.A) is irreducible for all ,~ > S(A).

1067

2.8.7. [178] An irreducible positiv e analytic OS is strictly irreducible. 2.8.8. [146, 178]. Assume that an OS exp(tA) is positive and analytic in a Banach structure E. If 0~. x*EE* are such that (exp(toA)x,x*) = 0 for some t o > 0, then

'i exp(tA)x, x* ) = 0

for

O~xcE and

any

Definition 21. We say that a functional p(x) defined on a vector space E is sublinear if (i) p(x+y)=p(x)'+p(y)

for x, yEE;

(ii) p(~.x)=~.p(x) for xeE, X~0. Definition 22. For a partially ordered Banach space E we define the canonical functional with respect to the given partial ordering to be the functional

p+(x):~inf{llx+y[l: y6E+}.

(8.1)

2.8.9. [178] Let E be a partially ordered Banach space. Then the canonical functional is sublinear and p+(x) + p+(--x) > 0. If E is a Banach structure, then p+(x) = [1x + It. Definition 23. Assume that in a Banach space E the functional p(x) is sublinear and continuous. An operator A L(E) is said to be o-dissipative if p(x) _< p(x - t a x ) for all t >_ 0 and x E D(A). Definition 24. Let p be a sublinear functional. A mapping T ~ L(E) is said to be a p-contraction if p(Tx) _ 0 if and only if its generator is p+-dissipative. 2.8.12. [178] Let E§ be normal. In order for a linear operator A to be the generator of a P+-contraction OS it is necessary and sufficient for the following conditions to be satisfied: (i) A is densely defined; (ii) A is P+-dissipative; (iii) ,~(M--A)~E for some )~ > 0. Definition 25. An operator norm in B(E) is said to be positively attained if

Ilnll=sup{llnxll:

0

x6E;

Ilxtl~l)

for every positive operator B E B(E). 2.8.13. [178] Assume that a cone E+ is sharp and the norm in B(E) is positively attained. In order for A to be a generator of a positive contraction semigroup it is necessary and sufficient that (i) A be densely defined; (ii) A be P+-dissipative; (iii) ~ ( M - - A ) =E for some A > 0. 2.18.14. [178] In a Banach (i) A is the generator of a (ii) A is densely defined, 2.8.15. [146, 178] Assume AOff(M, ~)

structure E the following statements are equivalent: positive contraction semigroup; ~(gI--A) =E for some A > 0, and A is p+-dissipative. that a cone E+ is sharp, and the norm in B(E) is positively attained. Then an operator

and the OS exp(tA) are positive if and only if

p§ for all x E E, n ~ N. 1068

A6$'(E), (o~, oo)6p(A) , and

Ap x]~<

M

0.-- o) ~

p. (x)

(8.2)

2.8.16. [146, 178] For an arbitrary Banach structure Proposition 6.15 remains valid when (8.2) is replaced by

11[R(L, A)" xl + !t 4

~0.--~o) n~ I; x+ ]1

(8.3)

for all x E E, n E N. Definition 26. An operator AE,~(M, o~) that is defined on a Banach structure E satisfies the positive minimum principle if, for all x ~ D(A)§ and q~E~ .-="E.-"f/ E | the equation (x,~p) = 0 implies the inequality (Ax,~o) _> 0. 2.8.17. [178, 206] The generator of a positive OS satisfies the positive m i n i m u m principle. 2.8.18. [162] Let A~?~'(hl,~o) and let there exist a core .~0 of operator A such that )c6g~0 implies [x[~ The satisfaction of the positive minimum principle for A implies that the OS exp(tA) is positiVe. 2.8.19. [121] Let A be the Laplace operator. Then for any f E LlI~ a) such that A f E LlI~ d) w e h a v e ReIsign f. Af l e A Ill.

(8.4)

Definition 27. For each x E E we define the linear operator sign x by the properties: (i) sign x x = kt (ii) [ s i g n x y f < lY[- for all y E E; (iii) sign x y = 0 if inf{[xl,h,l} = 0. 2.8.20. [126, 178] In a a - o r d e r complete structure, relative to any x E E the operator sign x is uniquely defined. 2.8.21. [121, 363] Let AE,~ (M, 0). Then the following conditions are equivalent: (i) the OS exp(tA) is positive; (ii) for all x E D(A) and all O~ifiD(A | (q~, sign~,,IAx)** ) ..< ( i x [ , A%o ) for each ), > 0 there exists a set K~ c D(A~ there exists a 0~-qED(A S) such that

such that (M -- A ~

separates E+\(0) from 0, i.e., for e a c h

O<xEE

<x,(~d--A~)~ > >0. 2.8.22. [363] Assume that under the conditions of 8.21 E is also Dedekind complete (i.e., any majorizable subset S c E has (sup S) ~ E). Then the following conditions are equivalent: (i) the OS exp(tA) is positive; (ii) for each x ~ D(A) and 05q~ED(A)*) we have <sign.~(Ax), qD)~< [x[, A*qD>, and for each ,k > 0 there exists a set K;~ E D(A*)+ such that (),I -- A*)KA separates E+\(0} f r o m 0. Definition 28. A linear operator A in a a-order complete Banach structure satisfies Kato's inequality if <sign:~Ax, t p ) ~ < l x [ , A*q~) for all x ~ D(A) and

(8.5)

0_
2.8.23. [120, 121] The generator of a positive OS in a a-order complete structure satisfies the Kato inequality (8.5). In order to state sufficient conditions we introduce the following additional constructions. Definition 29. A set S C E+ ~ is said to be strictly positive if for any x ~ E , it follows f r o m (x,~a) = 0 (~ E S) that X=0.

Ax ~ . x

Definition 30. A function ~ is said to be strictly positive if the set {c~ : a ~ 1~+) is strictly positive. Definition 31. An.element x ~ E is said to be a positive subvector of an operator A ~ L(E) if O~xED(A) and for some ,~ E R. 1069

2.8.24. [178] If exp(tA) is a positive OS in a Banach structure E, there exists a strictly positive subset K c A consisting of positive subvectors of A. 2.8.25. [178] If there is a strictly positive functional on E, then A* has a positive subvector. 2.8.26. [178] Let exp(tA) be an OS in a a-order complete Banach structure. Then the following conditions are equivalent: (i) the OS exp(tA) is positive; (ii) the operator A satisfies the Kato inequality and A* has a strictly positive set of positive subvectors. 2.8.27. [178] If there is a strictly positive functional defined on E under the assumptions of 8.26, then positiveness of the OS exp(tA) is equivalent to the existence of a subvector ~oof A* for which inequality (8.5) issatisfied. Definition 32. A linear operator has positive outer diagonal property if (x,~o) = 0 for O~x6D(A), 0 ~ep~E* , implies that (Axdo) _>0. 2.8.28. [125, 146] Assume that the cone E+ is solid. Then an OS exp(tA) is positive if and only if A has the positive outer diagonal property and R (~, A ) 9 0 for sufficiently large real A. 2.8.29. [125, 146] For a space with a normal solid cone and a densely defined operator A6,~(M, co) the following conditions are equivalent: (i) A is the generator of a positive OS; (ii) A has the positive outer diagonal property and ~(~,I--A) m E for sufficiently large A E R; (iii) R(X, A ) ~ O for sufficiently large A ~ R. 2.8.30. [125, 146] Conditions (i)-(iii) of 8.29 imply S(A) = ~o(A) = inf{A E R : Ax >_ Ax for some x E'D(A) n int E+}, where S(A) is the spectral bound of an OS. 2.8.31. [178] If an OS exp(tA) is positive in a Banach structure L1(fl,#) or L2(fL#) then s(A) = w(A). 2.8.32. [243, 244] Assume that exp(tA) is a positive OS in a Banach structure E. Then (i) if o(A) ~ ~, then S(A) is a point of the spectrum; (ii) If S(A) is a pole of order m of the resolvent R(A,A) and A is another pole with Re A = S(A), then its order may not exceed m; (iii) all of the numbers S(A) + ik 9 ~ (k ~ Z ) appear in o(A); (iv) if exp(tA) become compact, there exists an e > 0 such that in the strip S(A) -- ~ < R e A < S(A) there are no points of the spectrum of A; (v) if exp(tA) is irreducible and S(A) is a pole of the resolvent, then S(A) is a characteristic number of algebraic multiplicity one and the corresponding characteristic vector is a quasi-interior element; (vi) the spectral bound Ba(A) (i.e., the part of a(A) on the line S(A) + i~, where ~ ~ R) under the assumptions of (v) consists of characteristic numbers of algebraic multiplicity one of the form S(A) + irk with fixed v _>0 and k Z. 2.8.33. [178] If S(A) < 0 under the assumptions of 8.32, then there exists an co > 0 such that for any x o E D(A) we have Itexp (tA) Xot[<~Moexp(--or), where Mo may depend on x o. 2.8.34. If an OS exp(tA) is positive in a Banach structure E, then S(A) = oJl(A ) (see 2.7). Definition 33. A positive OS exp(tA) majorizes (or dominates) an OS exp(tG) if [ exp (tG) x t < exp (tA) Sx l for all x ~ E, t >__0. 2.8.35. [126] An OS exp(tG) is majorized by an OS exp(tA) if and only if IR(Z, G)xl
1070

and for each )~ > 0 there exists a set KA C D(A~

) for a l l

xED(G),

such that (x,(AI -- A~

(8.6) _> 0 for all ~o ~ KA and x ~ 0 .

2.8.37. [126] In case the space of 8.36 is Dedekind complete, Proposition 8.36 chan~es the same way as 8.21 and

8.22. 2.8.38. [121] Assume that exp(tA) is a positive OS, G is a densely defined linear operator, and (8.6) is satisfied. Then G admits the closure G. 2.8.39. [121] Assume that the conditions of 8.38 are satisfied. Then, if (AI - - G)D(G) is dense in E for some A > max{0.S(A)}, then G generates an OS exp(tG) that is dominated by the OS exp(tA). 2.8.40. [126] Given positive semigroups exp(tA) and exp(tG), then the conditions (i) exp(tG)~exp(tA)(/~0); and (ii) ( Gx, q~ ) -~ ( x, A~q0 ~ for all x ~ D(G)+, ~ E D(AO)+ are equivalent and imply satisfaction of the relation (iii) Gx ~Ax for O~.xED(A)f3D(G) 2.8.41. [126] If, under the conditions of 8.40, we have D(A) c D(G) or D(G) c D(A), then (i) follows from (iii). 2.8.42. [178] Let E be a Banach structure with order continuous norm, and let exp(tA) be a positive OS defined on E. Assume that we are given a semigroup lJ(t) such that the t'unction I[l(t)x is strongly measurable for every x E E and exp(tA) < lA(t), t _> 0, where sup It U(/)][ < o~. Then (J(t) is a positive OS. 0~t~1

Definition 34. By the modulus of an OS exp(tA) (denoted by lexp(tA)]) we mean the minimum positive OS that majorizes the OS exp(tA). 2.8.43. There are examples of semigroups for which there are no moduli. 2.8.44. [147] If A is a bounded operator in an order complete complex Banach structure that can be represented by the difference of two positive operators, then a modulus exists for the semigroup exp(tA). 2.8.45. [147] If for some OS exp(tA) in an order complete Banach structure there exists a majorizing OS, then the modulus [exp(tA)[ also exists. 2.8.46. [147] Assume that exp(tA) is an OS in an order complete Banach structure, and Q E B(E) is such that Q < "tl for some "t > 0. Then for the semigroups exp(tA) and exp(t(A+Q)) moduli either simultaneously exist or do not exist, and the generators A n and (A + Q)H are related by the expression

(A+Q)~=A~+Re (2. Definition 35. A linear operator A defined in a partially ordered space E with normal reproducing cone E is said to be resolvent positive if (i) (~0, o o ) c p ( A ) for sore& ~_0~R; (ii) R(;k, A)~O for a l l k>~0.

Definition 36. We denote the infimum of ~'o from Def. 35 by S~.(A). 2.8.47. [122] For resolvent positive operators with S+(A) < A < #, we have O~R(I~, A)~R()~, A). 2,8.48. [122] There are examples of resolvent positive operators that are not generators of OS's. 2.8.49. [1 :~] ~ If exp(tA) is a positive OS, then S,(A) -- S(A). Definition 37. A set Q c E+ is said to be cofinal in E+ if for each x ~ E+ there exists a majorant x_< y E Q . 2.8.50. [122] If A is a densely defined resolvent positive operator, D(A)+ is cofinal in E+, or D(A~ is cofinal in E . ~, where S(A) = w(A). 2.8.51. [122] Assume that A is densely defined and resolvent positive. Then, if I] R(Ao,A)x [[ >- c i[ x [], (x ~ E+) for some Ao > S(A) and c > 0, then A generates a positive OS and S(A) = ~'(A). 2,8.52. [122] Assume that the norm of E is additive on the cone E+ and D ~ ) ~ E . Then the following statements are equivalent: (i) A generates a positive group of operators: (ii) A and - - A are resolvent positive and there exist A Max{S(A),S('A)} and c > 0 such that [[ R(A+_A)x [[ > c I[~x If.. for all x E E+.

1071

2.9. Functions of the Generating Operator of an OS. We consider the class continuous on 1~§ positive when r > 0, and representable in the form

of functions ~(r) that are

~7

q~(r)----- l ra~(s)

(9.~)

0

where a is a measure that is positive on [0,oo) and such that 9 d~.(s)

(9.2)

0

Definition 1. Let A be the generator of a contraction OS. Then for x ~ D(A) we set [85, 252]

~(A) x - - \ A(A + Zl) -~ xdo(k).

(9.3)

We call ~o(A) the closure of operator (8.3). 2.9.1. [22] The operator --~o(A) is the generating operator for a contraction OS et&(t), for which

J

ql~(t)----- 3 exp (),A)d,u~ (~,). t > O , 0

where #t is a family of probability measures on [0,o~) (/~o = 50,/zt+s --- at */~s, Pt -- 60 weakly as t ~ 0,/~(R.) integral converges in the strong sense. 2.9.2. [22] The measure #t can be found from the scalar representation

t). The

e-t~ = f e-r;'dYt ('t.) 0

obtained from Bernshtein's theorem on completely monotonic functions. 2.9.3. [22] If aL/(t) is a bounded analytic semigroup, then so is aZ&(t)Examples of the function ~o. We represent the function ~o(r) --- r ~ (0 < c~ 1) in the form

~

sin(n~) i ~ r

S ~-lds"

(9.4)

0

Then from (9.4) we can obtain a definition of fractional powers of a generating operator:

0

For the function ~o(r) = In(1 + r) we have

In (1 + r ) ~

r 7T-Tds/s I

A(A+'t.l)-td)~/~..

and l n ( l + A ) = 1

2.10. Boundary Values of Operator Subgroups. Let 02/(0 ( E > 0 ) be a strongly continuous semigroup of bounded operators in a Banach space E, and let --A be its generator. In general, strong continuity of the semigroup at zero and denseness of D(A) in E are not assumed. Also, assume that the following conditions are satisfied:

a) . r

1072

={0};

b) the semigroup is a contraction semigroup: t1~ c) for any x E E the function o21 (t)x is differentiable when x E E. In virtue of these conditions, the function v(t) : = e / / ( t ) x with x E E satisfies the equation o'(t) = - - A v ( t ) ,

t>O.

(I0.1)

It turns out that this does not exhaust the functions satisfying (10.1). We introduce the norm tlxll-, : = Ir~(t)xll<~llxll into the space E, and denote the completion relative to this norm by E _ t. 2.10.1. [22]. The operator 02/(t) can be extended by continuity to a contraction operator ~ 2.10.2. [22] When 0 < s < t, we have the dense imbedding E _ s __C_E _ v The space E_: = p r lira E_t

in E _ t. is a locally

t.-~0

convex linear topological space. Definition 1. A semigroup H(t) (t > 0) defined in a locally convex linear topological space is said to be equicontinuous if, for any continuous seminorm p, there exists a continuous seminorm q such that for all t > 0 and xEE

p(ft(t)x)~q(x) 2.10.3. [22] The semigroup oS(t) is equicontinuous in E_. 2.10.4. [22] For any x ~ E the function ~ satisfies Eq. (10.1) and, conversely, for any function v(t) that satisfies (10.1) there exists an x ~ E such that v(t) -~-~Zl,(t)x. 2.10.5. [22] Let E be adjoint to some Banach space F and for a particular t o > 0 assume the operator is adjoint to some operator that is bounded in F. Then x ~ E belongs to E if and only if sup 11o21(t)x II ~ o~.

~

0
2.10.6. [22]. If E is reflexive, then x ~ E_ belongs to E if and only if the function

I~

[] is bounded close

to zero. For the case of analytic semigroups it has been possible to establish a connection between the behavior of the function ~ ( t ) x and membership of bounded values of x in certain spaces intermediate between E and E_. They are constructed like the space E_, but the role of the operator A is played by a function ~(A), where ~o satisfies conditions (9.1) and (9.2). We denote the corresponding spaces by E_(~). 2.10.7. [22] Assume that a semigroup satisfies the conditions of 10.5 and, moreover, is bounded analytic. In order for an x E E_ to belong to E_(~o) it is necessary and sufficient that there exist constants # > 0 and C > 0 such that

l]~ ( t ) x tl ~
1073

3. COSINE AND SINE OPERATOR FUNCTIONS (COF'S AND SOF'S) t

Assume that in a Banach space E we are given a COF C(t,A) and the associated SOF

S (t, A):

= f C (s, A)ds 0

(for the definitions, see 1.4). Recall that the generating operator (more accurately, the second generating operator) of a COF C(t,A) is defined as the limit

Ax:--lim 2 (C(h, A)x--x) h-+O

on all x E E for which it exists. 3.0.1. [137] we define

A~x:=llm c(2h. A)I2C(h. A)+I h-~O

X.

h"

Then for x ~ D(A 1) n D(A) we have Ax = Alx. For a COF C(t,A) we can define the first generating operator

r

~ x : ----lira

C (~. A)x--x

h--0+

on those x for which the limit exists. 3.0.2. [381] For a COF C(t,A) we have

h

D(A)~_D(~)

and

~x=O

for all x ~ D(A).

3.1. Fundamental Properties of COF's and SOF's. 3.1.1. [381] The operators C(t,A), C(s,A), S(t,A), and S(s,A) commute for all t, s E R. 3.1.2. [318] An SOF S(t,A) is continuous in the uniform operator topology. 3.1.3. [381] For all t, s E R we have

C(t,A)=C(--t,A), S(--t,a)=--S(t,A), S(O,A)ffiO, (ii) S(t+s,A)+S(t--s,A)=2S(t,A)C(s,A), (iii) S(t+s,A)---S(t,A)C(s,A)+S(s,A)C(t,A), (iv) C(t+s,A)--C(t--s,A)=2AS(t,A)S(s,A); (v) C(2t,A)=2C(t,A)--I, C(t,A)~--AS(t,A)~=I; (vi) C( (n+l)t, A) =bol+btC(t, A) + . . . +b,~+tC n+l (t, A),

(i)

where b o +blx + ... + Bn+lxn+l is a Chebyshev polynomial of the first kind of order n + 1. 3.1.4. [381] For any COF C(t,A) there exist constants M >_ 1 and oJ >_ 0 such that for all t ~ R

lie(t, A)II~M ch (cot), t~R.

(l.l)

Definition 1. The lower bound of the numbers ca of (1.1) is called the type of the COF and is denoted by oJr 3.1.5. [307] There may be no smallest oJ satisfying (1.1) with an appropriate constant M w (i.e., generally speaking, the lower bound of(A) is not attained). 3.1.6. [209, 226, 381] Let IIC(t,A)II _< M cosh (o:t). Then A ~ ( M , o2), the OS exp(tA) can be analytically continued to the right half-plane, and

co

exp(tA)-----

I

0

1074

s ~

Ie-~C(s, A)ds, tEff+.

However, there are examples of analytic OS's whose generating operators do not generate COF's (see [329]). 3.1.7. [169] An operator A generates a uniformly bounded COF if and only if A6gd (3/2,0) and EXP (A) = E . 3.1.8. [398] For any x E E and t, s E R we have t

(I) y : = I S ( r , A ) x d r E D ( A )

and A y = C ( t , A ) x - - C ( s , A ) x ;

s t

s

(ii) z : = f l C ( r , A)C(g, A)xdTd;fiD(A)

and Az:(C(t.S,-s,A)--

0 0

- - C ( t - - s , A))x/2; (iii) S(t, A)xEEk t

3.1.9. [398] If an element x runs through all of E, the set of elements y of the form Y= l S(z, A)xd~

is dense

$

inE. t

3.1.10. [398] For any x ~ E we have

lira t-IS(t, A ) x = x t-,-0

and lira 2t-~ f S(x, A ) x d : = x . t~O

0

3.1.11. [398] If x ~ E 1, the for any t ~ R (i) C(t,A)x6E l, S(t,A)x6D(A) and C' (t,A)x=AS(t,A)x; (ii) lira A S ( r , A)x=O and S'(t, A ) = A S (t, A)x. ~0

3.1.12. [398] Let x E D(A). Then for all t E R (i) C(t,A)x6D(A) (ii) S(t,A)xOD(A)

and C"(t,A)x=AC(t,A)x=C(t,A)Ax; and S " (t,A)x=AS(t,A)x=S(t,A)Ax.

3.1.13. [396] For all t, s ~ R we have

(i) C(2t, A)=C(I,A)2+C" (t,A).S(t,A); (ii) C'(t,A)S(s,A)=C'(s,A)S(t,A); (iii) C(t+s,A)--C(t--s,A) = 2 C' (t, A)S(s,A). k

t

(iv) (C (t, A ) - I) I S (r, A) dr = (C (It, A)-- I) I S (r, A) dr l

0

0

(v) ( A - - ~ 9 i sh (L (t-- s)) C (s, A) ds = ~ (C (t, A)--ch (~t)) 0

3.1.14. [211] The domain of the generating operator of a COF C(t,A) coincides with E 2, and for each x ~ D(A)

Ax = lira C" (r, A) x.

(1.2)

"t'~0

Sometimes the generating operator of a COF is actually defined by (1.2). We will denote the set of COF generators with bound (1.1) by ~'(M, to). 3.1.15. [381] Let A, G6~(M, ~). Then, if D(A) c D(G) and Ax = Gx for x E D(A), then C(t,A) = C(t,G) for all t E R.

1075

THEOREM 1 ([186, 209, 381, 387, 406]). An operator Aft(E) constants M and to the resolvent R(,~,A) exists when Re ~ > to and I dn

generates a COF if and only if for some

",,,

"(Z__e)~+~,

hEN.

(I.3)

3.1.16. [381] Sometimes (1.3) is written in the form

}1 dn

~<'--~-'{ (ReL__e)n+, + (Re~+e),~+, ')

(1.4)

for all Re ,~ > to, n E N. In practice, it is difficult to verify (1.3) and (1.4), so it is of value to find other conditions that imply generation of COF's. THEOREM 2 ([57]). An operator A~f~ (E) generates a COF if and only if D (A) = E and there exist constants M, 6, and to such that M

IIR(LZ, A)II~<

eor all ReL>~,

(1.5)

and the following bound is uniformly satisfied with respect to r E (0,6):

(~.6)

e~'~ch(Lt ) ~R (~~, A)xdX ~<~ ( t )lI x il, o)~teo where ~(t) ~ C(R). 3.1.17. [77] In connection with bound (1.5), note that the condition

IIR(~o2, A ) I I < ~ ,

ReX>o~,

for some ~ > 0 implies that the spectrum o(A) is bounded. 3.1.18. [310] When A is a normal operator in a Hilbert space, then it generates a COF if the following condition is met by its spectrum: {z2 : Re z > } c p(A) for some w. 3.1.19. [396] For Re A we(A), we have A2 ~ p(A) and

(1) kR (Z,2, A)x~_ i 0

e-arC (t, A)xdt, x~E;

0o

(ii) R (~,~, A) x = S e-xtS 0

(t, A) xdt, x~.E.

3.1.20. [307] For any At]~(M, oo) and x ~ E we have limll~,2R( k2, 3.1.21. [211] If x E D(AS), y E D(A), and o) > toe(A), then {1)

1076

t2

t'

1

C (t, A) x=x-}-~i Axq---~ A2x q - ~ I

A)x--xll=O.

eXtk-3R(z'~' e--too

AlA~xdk;

1

o~+too

(ii) C(t, A)y----ffn7 I e~%R(L2, A)ydL (0--lop

By writing the inverse Laplace transform in other forms we can obtain other analogous representations of C(t,A) and S(t,A). 3.1.22. [328] Let x e D(A k) for some k e N. T h e n for t e R we have Taylor's f o r m u l a (for the analog for OS's

see 2. I ): ~2n--2

C(I, A ) x = x + ~q-Ax+ ... + (2.--2)t An)c-}t

+ I (t--s)~=-' C (s, A) A=xds. (2n-- 1)I 0

3.1.23. [331] For any x E 1~o (see (2.1.16),

A~f~(M, to) , and t E R, we have

C (t, A ) x = vr t~A,xl(2k)!, k=0

and for each x ~ 1~o the function t ~ C(t,A)~ can be continued to a function that is analytic in the entire complex plane. 3.1.24. [209] If x E E,

t

(1.7)

where c o n v e r g e n c e is u n i f o r m with respect to t f r o m any c o m p a c t set in R\(0). d/t

3.1.25. [416] The expression ~(X, k):----a-~(XR(X2, A)) in (1.7) can be written in the form

(i) ~ ( k , k)-----.~l (X~+~+ c~+~Xk-~A+ ... +c~+lXA k/2) X X (k2I-- A) -(k+l), tz is even; k+l~ / .

(1i) ~'/(X, k ) = --k! \~.k+~+c~+~Xk-~A + ... + A T } ( X ~ I _ A) -r rt i s

odd;

k

(iii)

~2(k, k)= X (--1) j (k+ 1)t j! ~,(2X)-'J-k (X~I~A)_(I+~) /=k/2

( k - - j ) t (27--k + 1)t

/Z is e v e n ; fi

(iv) ~(X,k)=

~_~ (--I)/ (k+OQfZ(~/-k (~--j)I (2j--k + I)~

j=~--_l+l

rt i s

(X#_A)_(j+I),

odd.

3.1.26. [142, 373] For a C O F C(t,A) an SOF S(t,A), and any x ~ E and t ~ R we have k

(i)

1

C (t, Z)x---lim ~_oj~oC~tC[(--1)l-][I--(t/2k)2A]-(2k-'+l'x; 1077

n

#

(ii) C ( t , A ) x = = l t m Z Z ~2k n~

I-[

[

4=0

t

-).

1=0

\2,'l--I2n+l--k+/l

x L --~2--~-r) ~J oo

(iil~

rrt

x;

k

C(t, A)x-~limexp(--nl)m~= ~ ~ ~ (ntym "~'~,,1, nt + l '(I --tt-~A) -~] (I X [I + 2m--'2k n--I

(iv)

S(t,

m

n-2A)-(~"-~+l)x;

k

A)x-----lirn t n-.,,

~

C]

-~

"" ra=O k--O 1 ~ 0

where convergence in all cases is uniform with respect to t e o Y ~ R . 3.1.27. [373] Under the conditions of 1.26 we have

C (t, A )x=lim ~ ~ ~ r~m"~,,~ C~( -- 1)k-Jt2m(1--t)zn-gm-tX n~eo

m=O k=O ]=0

L - - t)-l-'2m---"---'~-~ * 2n--2m 1 t ( I - - ( 2 n F 2 A ) -1 ] • r(l

(I --(2n)-2A)-(2m-k+/)X;

uniformly with respect to t ~ [0,1]. Other relationships of this type are given in [373]. Notation:

St(A):= ~p(A): =

IIZ~xll-~<

xEE:

xEE:

oo

I!A~xil't2k/(2k) ! <

are

$tieltje-~

vectors,

~ eor sore

t>0

~!jA~cll<~ f o r a l l

t>0.

are semi-analytic vectors,

9pp(A):=xeE: k=0

3.1.28. 3.1.29. 3.1.30. dense in E. 3.1.31. 3.1.32.

"

"

[168] Let At?~(M, to) and l~o be constructed for a given OS exp(t,A). Then /~0c~p(A) [168] ~(A)c~p(A)c_St(A). [168] Let ~pp(A)=E. Then the set of vectors x of D(A ~ ) with the property that II Akx [168] Let [168] Let

AtS~(/tl, to).. Then ~(A)fl~pp(A)=E. ~pp(A) = E and assume there exists an operator

G6~'(E)

II 1/k __o(k)

is

such that

(i) G -1 E B(E); (ii) G 2 = A, where the operators __.Gare dissipative. Then

1078

AE$'(M, to) and C(t,A) = (exp(tG) + exp(--tG) + exp(--tG))/2, where G e l ~ ( l , 0 ) . 3.1.33. [168]. Assume that A 1 is closed, that St(A1) is total in E, A2 ~?~'(M, to), and A 1 C A 2. Then A~ = A z.

3.1.34. [168] Let A be a closed symmetric and semibounded operator in a Hilbert space H. Then A is self-adjoint if and only if the set St(A) is total in H. We define the following sets: NBI o = { t > 0 : the operator C(t,A) does not have a b o u n d e d inverse), NBI 1 = {t > 0 9 the operator S(t,A) does not have a b o u n d e d inverse}. 3.1.35. [250] Assume that the operator C(t,A) -- I is c o m p a c t for all t ~ R. The sets NBI o and NBI 1 are simultaneously either e m p t y or have cardinality of a c o n t i n u u m and there exist constants %, % > 0 such that NBIj c (r162 j = 0.1. 3.2. Reduction of the C a u e h y Problem for Second-order Equations to the C a u c h y Problem for First-order Equalions. In a Banach space E we consider u n i f o r m l y well-posed C a u c h y problem (1.1.)-(1.2) with n = 2 (t E R):

u"(tj=Au(t), u(O) = u 0, u ' ( O ) = u L

Wo e i.et,oo.er ,or, ~r

= (y, Ax) with domain

(OA

to. ,on,,eolemont, ,,.

(2.t)

or

.,tot,erorm l.

D ( ~ ) = D ( A ) X E 1.

3 . 2 . I . [27t] It is clear that D(A) C E 1. 3.2.2. [271] The set E 1 is dense in E. T H E O R E M 1. [271] The space E 1 with norm

ilx ile,=il xtl-+- 0~<1<.1 sup ii C'(t, A)xll is a Banach space. The operator

.9r

(2.2)

generates the group of operators

IC(t,A)

S(t,A)

e.xp(t3C)(x, y): = ~ A S ( t , A) C(t, A)) (x, Y)=

= (C(t,A)x+S(t,A)y, A S ( t , A ) x + C ( t , A ) y ) in the Banach space E 1 x E. 3.2.3. [413,414]. Assume we are given a C O F C(t,A). Then E 1 coincides with the closure o f D(A) in the norm

i!x . : = , i x j i + s u p:>,., h ! ( z1_ o ~ ) . + hEN 3.2.4. [271] The resolvent of the operator

(~,I--,~r

',~

~ ~-~ an A(zZ-- A )-lx I/.

is of the f o r m

I'k( " ~ I - A) -t (L2I--A) -~ '1 92 - A)-I k(k21_A)-tj f o r t, Ep(,~,).

(2.3)

3.2.5. [271] If u(t) is a solution to problem (2.1 and v(t) := u'(t), then the vector (u,v) T is the solution of the uniformly well-posed C a u c h y problem

1079

in the Banach space E 1 x E. Assume that the Banach space I~1 is continuously and densely imbedded in the Banach space E, where D(,~) c_ 1~1 for some .~ c L(E). Then the Cauchy problem (t ~ R)

(:),<,>_(o is uniformly well-posed in the space l~ 1 x E and

(2.5)

p ( . ~ ) 4 =~ .

3.2.6. [271] The OS corresponding to problem (2.5) can be represented in the form

exp (t,~):

[G~l(t)

=tO.oz(t)

Gi2(t)]

G22(t)J' t>~O,

(2.6)

where the family G22(t) is the COF C(t,A) and coincides with G l l ( t ) on E 1. 3.2.7. [271 ] For x E E and y E l~1 we have

G12(t)x=S(t,A)x and G2,(t)y=C'(t,A)y. 3.2.8. [271] The spaces I~1 and E 1 of Proposition 2.3 coincide up to equivalence of norms. We should note, however, that investigation of problem (2.1) by reduction to a system is extremely awkward, since the space E 1 is defined either in terms of a COF C(t,A) or as a power of the resolvent. As a rule, we have information only about the operator A. Thus, other methods of reducing problem (2.1) to a system are of interest. 3.2.9. [398] Assume we are given a Hilbert space E and a self-adjoint negative definite operator A. Then A ~ ' (M, o) and the corresponding space E 1 coincides with D(AU2). Assume that a uniformly well-posed problem (2.1) is of the form

u"=a2u; u(0)=u ~ u'(0)=u~,

(2.8)

where

GEm'(E) . Definition 1. A solution u(t) of problem (2.8) is said to satisfy condition (K) if u' (t)6C ([0, T], ~ ) ( G ) ) . 3.2.10. [59] Problem (2.8) has a unique solution satisfying condition (K) if and only if the following Cauchy problem is uniformly well-posed:

,,,)<

(.) ,o> __-

(2.9)

The analog of condition (K) that allows us to simplify investigation of problem (2.1) by means of OS's is condition (F): Definition 2. A COF C(t,A) satisfies condition (F) if the following conditions are satisfied: (i) there exists an operator G6~(E) such that G ~ = A and G commutes with any operator of B(E) that commutes with A; (ii) the SOF S(t,A) maps E into D(G) for all t ~ R; (iii) the function GS(t,A) is continuous in t ~ R for any fixed x ~ E. 3.2.11. [211] When condition (F) is satisfied for every t E R we have GS(t,A) E B(E) and D(kG) c_ E 1. 3.2.12. [211] There exist a Banach space E and a uniformly bounded COF C(t,A) such that condition (F) is not satisfied. 3.2.13. [211] We can always use A b := A - - b 9"9 I for b > we(A) to construct A b and G b such that Gb 2 = A b and G b c o m m u t e s with any operator of B(E) that commutes with A b.

1080

3.2.14. [210] The operator G b of 2.13 can be constructed, for example, thus: er

Obx: =--i.~ I "t~-I"~O'I--A~)-1(--Asx)dX" 0

T H E O R E M 2. [396] Let A and G be operators satisfying condition (i) of Definition 2, and assume that 0 E p(G). The following conditions are equivalent: (i) the COF C(t,A) satisfies condition (F); (ii) the operator G generates the OS exp(tG) on E, (iii) the operator

(iv) the operator where

O O) t7 O

.~/:=

with domain D(A) x D(G) generates a Co-group on E x E;

(%) with domain D(A) x D(G) generates the Co-group

exp(t,sr

on

~(G)XE

,

!~9 (G) is the Banach space D(G) with the graphic norm; (v) we have D(G) _ El; (vi) D(G) = E 1.

3.2.15. [21 i] Let AECg(M, 0) and assume that E = H is a Hilbert space. 3.2.16. [414] The following condition is equivalent to conditions (i)-(vi) D(G) is dense in E and there exist constants M > 0 and w _> 0 such that functions A(AgI -- A) -1 and G(A2I -- A) -1 are strongly differentiable an infinite n e N v {0}

[I

d

l / ( x - nl|

Then condition (F) is satisfied. of Theorem 2: A2 E p(A) for all A > w, the operator number of times when A > to, and for

?1

(~-Z)"(O(k~l-- A)-I) [ 4 M "

3.2.17. [396] Under the conditions of Theorem ~ " we have

exp(tG)=C(t,A)+GS(t,A), C (t, A) = (exp (tG) +exp (--tG) ) /2; (i)

,'G~I

(iii) exp (t.-Qr (x, v) = AS(t, A)x+C(t, A)V),

(C(t,A)x+S(t,A)y, (x, y)~D(G)NE, teR.

3.3. Spectral Properties in the Theory of Cosine Operator Functions. As in the case of operator semigroups, necessary and sufficient conditions for A to generate a COF can be stated in terms of conditions on the spectrum and bounds for the resolvent - - s e e 3.1. 3.3.1. [327] Assume that we are given a COF C(t,A). Then

(i) ch (t I ~-(X))g~r(c(t, A));

(t I/ P~(A))-----P(r(C (t. A)); ch (t I .?(I(A))~R~(C (t, A)).

(ii) ch (iii)

(3.1)

1081

3.3.2. [328, 4] If # E Rcr(C(t,A)) and (An}n~N is the set of roots of the equation # = exp(A n . t). then ),fi =~ Rcr(A~ for some fi and ,kn2 $ Pa(A) for any n E N and # ~ Pe(C(t,A)'). 3.3.3. [327, 4] If # E Ce(C(t,A)) and An are as in Proposition 3.2. then .kn2 ~ Ca(A) ~, p(A). It is possible that .kn2 (A) for all n ~ Z in some cases. 3.3.4. [215,280] If E = H is a Hilbert space and AE~(M, 0) or C(t,A) is a family of normal operators, then

cr(c (t, A))=ch (t 1 or(A)), t~R. 3.3.59 [167] Let a COF C(t,A) satisfy condition (F) and assume that E = H is a Hilbert space. Then # 6 p(C(t.A)) if and only if {z9 : cosh(zt) = #} _ p(A) and sup{ ~11zR(z,A) :;] : cosh(zt) = #} < pc. 3.3.6. [306] Let AE~'(M, 0). Then (i) o(A) C R ; (ii) if E ~ {0}, then a(A) ~: ~; (iii) the spectrum a(A) is bounded if and only if A ~ B(E). 3.3.7.[13]For

AE~(M, r

and the corresponding matrix operator 3 r

O / ) , that appears in the reduction

of Cauchy problem (3.2.1) and is given in 3.2 we have

('4 : ~ , e o ( , ~ ) } = , ( A ) . 3.3.8. [4] There exist a COF C(t,A) and a Banach space E such that the sets r~ : = { t . 0E9(C (t. and r3:={t:OECo(C(t,A))} are dense in R, w h e r e R = r l u r 2 u r 3.

A))},

r2 : =

{t:OePo(C(t,A))},

3.3.9. [133] For a COF

C(t. A ) = ~ =~ r-'kA~ (2~)!' A t ~

(i) O~
, defined in a Banach algebra with identity, we have

~ ; 3r

(ii)) 9

A)--lexp(--;,.t)C(t,

A)dt for ReL>coc(A);

0

(iii) R(k ~, A)----Iexp(--'~t)S(t,

Aldt for Re~.>o~(A);

0

(iv) C(t, A)=.Z~@I exp(~,t)~d~(Lz,A)dL, tER; Y

(v) S (t, A ) = ~

l exp (~.t)R (~2, A) d;,., tER, -f

where 7 is some curve that includes the spectrum of AE~ . 3.3.10. [133] Under the assumptions of Proposition 3.9 we have

oc(AF=

sup (IT.i+Re~.)/2. ~.~cd, A J

3.4. Supplementary Comments. The foundations of the t h e o r y o f COF's were laid in[280-286. 208-216.381. 396-399]. For more on reduction o f t h e problem u"+ Bu'+ Au = 0. u ( 0 ) = u ~ ul. to a system ofequations, see[164. 254-256].

1082

The case of p(A) = ~ for problem (2.1) was considered in [389]. The construction of solution families is analogous to COF's, but the existence o f the resolvent (A9 9 I -- A) - 1 is assumed on some Y c E with a norm stronger than the original. A d i f f e r e n t a p p r o a c h to definition o f the sine and cosine operators was taken in [56]. An attempt to define a C O F in terms of easily verified properties of the operator A has led (see [238]) to an u n b o u n d e d COF. It was shown in [349] that D(Ab~/2+')~E~D(Abl/2-'), where A b := A - - bI with appropriate b and ~ > 0. The properties of ~v are used in automatic control theory -- see [400]. A saturation theorem for COF's was proved in [201 ]. Linear affine C O F ' s were discussed in [234-235].

4. OPERATOR SEMIGROUPS AND COSINE OPERATOR FUNCTIONS THAT ARE C O N T I N U O U S IN T H E U N I F O R M O P E R A T O R T O P O L O G Y

4.1. Operator Semigroups that are Continuous in the Uniform Operator Topology. r

4.1.1. For any b o u n d e d operator B E B(E) the series

X(tB)~/M

converges u n i f o r m l y and the operator

k~0

function

exp(tB)= ~ (tB)~/k!, t>~O,

(1.1)

is an OS with generator B. Definition 1. A n OS exp(tA) is said to be continuous in the u n i f o r m operator topology ( c o n t i n u o u s w i t h respect to norm) beginning at t o (sometimes we will say "becomes continuous with respect to norm") if there exists a t o > 0 such that the function t ---, exp(tA) f r o m [t o,C~) into B(E) is continuous in the u n i f o r m operator topology. We say that an OS is continuous with respect to norm if t o = 0. 4.1.2. If B ~ B(E), then the OS exp(tB) is continuous with respect to norm. T H E O R E M 1. [35] An OS exp(tA) is continuous with respect to norm if and only if A E B(E). 4.1.3.[41]If Afi~(M,o~)

a n d D ( A ) is a set of the second category in E, then

limi!exp(rA)~l.i=0

and

'~0 ~

A s B(E). 4.1.4. [101] If t~,, o _ D(A) for some %, then A exp(tA) E B(E) for t >_a o and the OS exp(tA) is continuous with respect to norm when t > %. 4.1.5. [341] If I ~ c_ D(A) for all a > 0, then the OS exp(tA) is n-times continuously differentiable in the uniform operator topology with t > 0 no matter what the n ~ N. 4.1.6. [ 126] If exp(tA) is a group of operators and

lira ~ exp ( r A ) - - Ili <~ 2 , then the generating operator A

B(E). 4.1.7.[341]IffZaCD(A)

foralla>0and

iimt~iAexp(tA)i.~
,thenA~B(E).

t~0+

4.1.8. [10] For a C o ' - s e m i g r o u p exp(tA) the following conditions are equivalent (the same holds for wcontinuous semigroups): (i) there exist ~ > 0 and 6 > 0 such that

tlexp(tA)--lil~l--e (ii)

for 0 ~ t < 6 ;

limliexp(tA)--Iii=O. t~O+

4.1.9. [231] If

A@gd(0, o~) and

m A ~ f f ( M , co) , then A ~ B(E).

1083

4.1.10. [101] If there exists an unbounded open connected set f~ that contains zero and numbers 0 _ 0 and exp(toA) E Bo(E) for some t o > 0. Then the operator exp(tA) is compact for all t > t o and the semigroup exp(tA) is continuous with respect to norm for t > t o. Note that here we have not assumed that the semigroup exp(tA) is a Co-semigroup (OS). Definition 2. A Banach space E is said to be a Grothendieck space if any sequence {Xn*}, Xn* ~ E*, that converges in the *-weak topology to zero converges to zero in a weak topology. Definition 3. We will say that a space E has the Dunford--Pettis property if for any other Banach space X each weakly compact operator T:E ~ X maps weakly con('ergent sequences into sequences th~it converge with respect to norm. 4.1.13. [300] An example of a Grothendieck space with the Dunford-Pettis p r o p e r t y is L ~176 4.1.14. [300] An OS exp(tA) defined on a Grothendieck space with the Dunford--Pettis property is continuous with respect to norm, i.e., A ~ B(E). 4.1.15. [300] Let an OS exp(tA) be given in a space E with the Dunford--Pettis property. If the family exp(tA)'" is an OS on E**, the OS exp(tA) is continuous with respect to norm. 4.1.16. [363] Let A E B(E), where E is a Banach structure. The following conditions are equivalent: (i) the OS exp(tA) is positive; (ii) ( ~, stgnx..(Ax)** ) ~< ( A ] x ! , ~ ) for all x ~ E and o_0; (iv) the operator A + ]]A I] " I > 0. 4.1.17. [363] If A, B E B(E), where E is a Banach structure, the the following conditions are equivalent: (i) lexp(tB)xl<exp(tA)lx[ for any x ~ E; (ii) ( A I x [, q~ ) > ( q~, sign~,, (Bx)**) (iii) A ~ B and A + B ~ - - I I A q - B ] [ ' I .

for all x 6 E and

0
4.1.18. [363] If A, B ~ B(E) and the operators C and D are invertible in B(H), where supl I l ~ C e x p ( ~ A ) tE~

•

D e x p (tB)il ~< 1 , then the operators A and B are similar. 4.2. Cosine Operator Functions that are Continuous in the Uniform Operator Topology. Definition 1. A COF C(t,A) is continuous in the uniform operator topology (continuous with respect to norm) if the function C(t,A) : R ---, B(E) is continuous in the operator norm. 4.2.1. Let A COF C(t,A) be continuous in the uniform operator topology. Then A ~ B(E) and

C(t, A)= ~ t2*Ak/(2k)l, t~R,

(2.1)

k=O

where the series converges uniformly with respect to t on each finite segment [0,T]. Sometimes, by analogy with the scalar case, series (2.1) is written as cosh(tA1/2). 4.2.2. [396] Let A E B(E). Then series (2.1) is a COF with generating operator A. T H E O R E M 2. [134] Each of the following conditions is equivalent to continuity with respect to norm for C(t,A): (i)

llm]lC(t,A)_l,i=O

;

t~0

(ii)

limiit-~S(t, A ) - - I I I _ _ - 0 ; t~0

(iii) the generating operator A is bounded; (iv) ~ ( C ( t , A ) ) c E l f o r a l l t ~ ( c ~ , f l ) for some c~ < 3; (v) the inclusion ~(S(t, A) ) c-D(A) and strong continuity of the function r ---, AS(t,A) occur for all t ~ (c~,3) for some a < 3. 4.2.3. [7] The generating operator A of a COF with a nonquasianalytic weight X is bounded if and only if one of the following conditions is satsified: (i) for some ~ > 0 we have SUPo
Ill

< 1;

(ii) the COF C(t,A) is the restriction to R of an entire function C : C ~ B(E) of exponential type (equal to r'A)). 4.2.4. [397] Assume that the COF C(t,A) is strongly twice differentiable at zero. Then A ~ B(E). 4.2.5. [306] Let AE~7(M, 0) . Then the operator A E B(E) if and only if the spectrum a(A) is bounded. 4.2.6. Let the COF C(t,A) be continuous with respect to norm. Then (i) ~1

I/S

I

(2/t 9) S(T,A)d'~--I = 0

; h I S(x, A)dr

(ii) for sufficiently small h the operator

has a bounded inverse;

0

(iii) for sufficiently small h we have

A=(C(ft, A)--I)

S(r,A)ar

.

4.2.7. [371] Any COF C(t,A) defined on a Grothendieck space with the Dunford--Pettis property is continuous with respect to norm, i.e., A E B(E). 4.2.8. [7] If B E B(E) and the COF C(t,B) is uniformly bounded with respect to t E R, we have the Bernshtein inequality

ii B [l ~ r (B). sup [i C(t, B)!r, ~ER

where r(B) is the spectral radius of the operator B. 4.2.9. [34,342] Let B E B(E). Then the functions C(t,--B2)x and S(t,--B2)y are uniformly bounded with respect to t ~ R for all x, y E E if and only if on E there exists an equivalent norm I]']l* such that [lexp(itB)][" < 1 for t ~ R (such operators are said to be Hermitian equivalent on E). 4.2.10. [34, 342] An operator B is Hermitian equivalent on E if and only if there exists a constant C > 0 such that Ilsin(tB) l[~C, 4.2.11. [133] Assume were are given, in a Banach algebra ~,2 > supaeo(a)(LkI + Re X)/2 we have

C(t, A)=~

I

t~R. -~ with identity, a COF

e;-:~R(ZL A)d;.,

t>10,

e;.'R(ZLA)dZ,

t>~O.

C(t,A), A6~1 . Then, for

v--i.~

9

S(t,A)=~

I

l V--.'r

4.3. Supplementary Comments. In [299] there is a discussion of the behavior of the spectrum a(T(t)) of an operator function T(t) that is a continuous function of t in the uniform operator topology. Conditions under with an OS exp(tA) is representable in the form exp(tA) = P exp(tB), where P, B E B(E), PB = BP and the operator I - - P is of finite rank, were considered in [300]. See also Propositions 2.6.13, 2.7.25, 2.7.47, and 8.1.4.

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5. ALMOST P E R I O D I C AND P E R I O D I C O P E R A T O R S E M I G R O U P S AND COSINE O P E R A T O R F U N C T I O N S Assume we are given a function f(t) : R ~ E. A number r ~ R iS s a i d t o be an e-period of the function f(t) if

IIf(t+v)--f(t) Ilk<e, t~R. We denote the set of all e-periods of the function f by J(f,e). D e f i n i t i o n 1. We say that a function f(t) is almost periodic (and write a.p.) if for any e > 0 the set J(f,e) is relatively dense in R. This means that for any e > 0 there exists a number l~ > 0 with the following property: any subinterval of length l in R intersects J(f,e), i.e.,

In, a+lJN1(f, e)=/=0

for a l t aeR.

5.0.1. For any a.p. function f(t) and any e > 0 the set of e-periods {r~} is closed. 5.0.2. Every periodic function is almost periodic. 5.0.3. [118] Assume that an almost periodic function f(t) : R --, E is continuous. The function f(t) is uniformly continuous on R. 5.0.4. [ l l 8 ] If f(t) is an a.p. function, then the set of values Rf := {x E E : x = f(t), t ~ R} is relatively compact in E. 5.0.5. [ l l 8 ] If f : R -* E is an a.p. function that has a uniformly continuous derivative f'(t) on R, then f' is an a.p. function. T H E O R E M 1. ([107,318]) Let f E C(R;E). Then f is an a.p. function if and only if the family of function {f(t + h)}_oo
5.0.6. [61] The sum f(t) + g(t) of two a.p, functions is an a.p. function. The product of an a.p. function f and a numerical a.p. function ~o(t) is an a.p. function; ~(t)f(t) : R ~ E. 5.0.7. [118] For every a.p. function f the following limit exists: T

a ( k , f):----lim(2T)-'r_,~ _ t f ( t ) e - i ~ t d t

for any

kER.

5.0.8. [118] Under the assumptions of 0.7 the following limit exists:

c+T

a(0,/):=lim(2T) -1 I f(Qdt T--*-oo

uniformly

with respect

t o c{~R.

r

D e f i n i t i o n 2. The function a(A,f) is called the Bohr transform of f. We call the numbers A for which a(A,f) # 0 the Fourier spectrum of F and denote them by SP(f). 5.0.9. [118] The Fourier spectrum of any a.p. function is no larger than countable. 5.0.10. [118] If, under the assumptions of 0.7, SP(f) does not contain a sequence converging to zero, the function t

I f(s)ds

is almost periodic.

0

D e f i n i t i o n 3. A continuous function f ~ C(R;E) is said to be recurrent if for any numbers e and T the set

L (e, T; f): = (TeR :sup {lf (t + ,)--f (t){{~< ~1 {tl~T

is relatively dense in R.

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Definition 4. A subset

~'cR

is said to be harmonious if for any ~ > 0 the set

relatively dense in R. Definition 5. A family of functions

~=

N {r:le~X~-- 11,,<~} zE~

is

{f} that maps R into E is said to be u n i f o r m l y almost periodic if the

set N i ~ ' / ( r , e) is relatively dense in R for each ~ > 0. 5.0.11. [144] If a set ~" is harmonious, there exists a number 6 > 0 such that the distance between arbitrary points of ~ is greater than 6. Definition 6. A function f : R ~ E is said to be weakly a.p. if (f(t),y*) is a.p. for all y* E E*. Weak periodicity is defined similarly. 5.1. Almost Periodic and Periodic Operator Semigroups. Definition 1. An OS exp(tA) is said to be an almost periodic semigroup (and we write a.p. OS) if for each x E E the function exp(tA)x : R - . E is an a.p. function. Definition 2. An OS exp(tA) is said to be uniformly almost periodic if the family of functions {exp(tA)x: x E E, li x 11= 1} is uniformly almost periodic. Weak almost periodicity for an OS is defined analogously by way of a.p. functions of the form <exp(tA)x,x') for allxE E,x*~E*. 5.1.1. [15] For an a.p. OS exp(tA) the space E can be factored into a direct sum E = Est @ Eap (see 2.7 for definitions). Definition 3. The subspaces Est and Eap are called the interior and the boundary, respectively. 5.1.2. [15] The subspaces Est Eap are invariant with respect to the OS exp(tA). 5.1.3. [144] An a.p. OS exp(tA) can always be imbedded in an a.p. group exp(t~,) as follows:

J'exp(tA), t > 0 ,

exp(tA)---- I,exp(--tA)-~, t < 0 . T H E O R E M 1. [144] A groups of operators exp(tA) is a.p. if and only if the following three conditions are satisfied: i) the group exp(tA) is uniformly bounded; ii) the spectrum a(A) _ iR; (iii) the set of linear combinations of the characteristic vectors of the generating operator A is dense in the space E. 5.1.4. [144] Let E be weakly sequentially complete (for example, in particular, E might be reflexive). In this case a group exp(tA) is an a.p. group if and only if it is weakly a.p. 5.1.5. [144] Let exp(tA) be a uniformly bounded group in a reflexive space E. If the function exp(tA)x is almost periodic for all xEgC(A) , then exp(tA) is an a.p. group. 5.1.6. [144] Assume that the group exp(tA) is almost periodic. Let it/(r/6 R) be an isolated point of the spectrum ~r(A). Then it/ is a simple pole of R(),,A), irl E Pa(A~, and the corresponding Riesz projector is given by the formula

g

P(iq}x=lim t -11 e<'~exp(sA)xds"

(1.1)

0

5.1.7. [I 5] Let A e,~ (M, 0) and assume that a(A) n iR is no larger than countable; also, for each characteristic value i# e a(A) (~z ~ R) of the operator A* and each functional l e a ~ ( i u l * - A * ) , f=/=O , assume there exists an element xe./F(igl--A), such that (x,f) ~ 0. Then the OS exp(tA) is a.p. 5.1.8. [15] Let A6~,(M, 0) and let the space E be reflexive. If e(A) n iR is no larger than countable, the OS exp(tA) is a.p. 5.1.9. [15] If an OS exp(tA) is a.p. then nul (it,d - - A ) = nul (igl*--A

*)

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for all # E R. In particular, ,h~(i,ul--A) = {0} if and only if ,4,~ -- {0} . 5.1.10. [144] If a group exp(tA) is almost periodic and Per(A) is bounded, then the generating operator A E B(E), i.e., is bounded. 5,1.11. [144] An a.p. OS (see 1.3) can be uniformly imbedded in an a.p. group. 5.1.12. [144] A group exp(tA) is uniformly almost periodic if and only if the following three conditions are satisfied: (i) the group exp(tA) is uniformly bounded; (ii) the set (1/i)e(A) is a harmonious subset of R; (iii) the set of linear combinations of the characteristic vectors of the generating operator A is dense in the space E. 5.1.13. [144] If exp(tA) is uniformly almost periodic, then e(A) consists of simple poles of the resolvents R(A,A) and a(A) = Pa(A). 5.1.14. [144] Let A E B(E). Inthis case the group exp(tA) is a uniformly a.p. group if and only if it is uniformly bounded and E is representable as the direct sum of a finite number of characteristic subspaces of the operator A. 5.1.15. [143] A uniformly b o u n d e d OS exp(tA) is periodic with period T if and only if the following two conditions are satisfied: (i) the set (T/2rri)e(A) _c Z, the integers; (ii) the set of linear combinations of characteristic vectors of the generating operator A is dense in E. 5.1.16. [143] Assume that the OS exp(tA) is periodic with period T. Then (i) the resolvent R(,~,A) is a meromorphic function with only first-order poles; (ii) T

,-9,(~,, A ) = ( 1 - - e - ~ - r ) -1 f e-;.'exp(sA)ds; 0

(iii) the orthogonal spectral projectors corresponding to the poles Ak are of the form T

1

P f f ' * ) = - f f e-~'k%xp(sAJds; 0

(iv) for any x~ E

A P ( , . , ) X = ' h k P Q , k)X and e x p ( t A ) P O ~ J x = e zkt p~,~)x. "" " 5.12.17. [143] The following statements are equivalent: (i) the OS exp(tA) is periodic; (ii) the OS exp(tA) is weakly periodic; (iii) the OS exp(tA) is periodic as an operator function. 5.1.18. [143] Assume that the function exp(tA)x : R --. E is periodic for every x E D(A). The OS exp(tA) is periodic. 5.1.19. [143] Assume that E = H and the OS exp(tA) is periodic. Then the following conditions are equivalent: (i) the OS exp(tA) is a contraction OS; (ii) the OS exp(tA) is unitary (see defr in 7.1); (iii) the OS exp(tA) is normal. If the OS exp(tA) is also self-adjoint, then exp(tA) - I for all t >_0. 5.1.20. [143] Let A6$r(M, to), and let P be a projector. Then exp(2,"rA) = P if and only if the following conditions are satisfied: (i) o(A) __ iZ; (ii) the spectrum o(A) consists of simple poles of the resolvent;

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(iii) there exists a constant M >_ 0 such that

II(I--P)R(LA)II~Mmax{1, e-2~R~;'},

Re ~4= 0;

(iv) the set of linear combinations of the characteristic vectors of the generating operator is dense in E. Definition 4. A function f(t) : R ~ E is said to be almost automorphic if for any sequence {O~n')nEl~l. there exists a subsequence {C~n}_C {an"} such that

limf(t-+-c~,,)=g(t) limg(t--~,,)=f(t)

for any tER; f o r any

/ER.

n,--), o ~

5.1.21. [427] Let B E B(E) and let u(t) be an almost automorphic solution of the equation u'(t) --- B(u(t), t E R. Then either i ~ ! t t t { t ) l ] > 0 , or u(t)-= 0 for t E R. 5.1.22. [428] Proposition 1.21 remains true if AEff(M, ~), and u(t) is an almost periodic solution. 5.1.23. [426] Proposition 1.22 remains true if the condition of almost periodicity is replaced by the requirement that the solution u(t) : R ~ D(A) be weakly almost periodic. 5.2. Almost Periodic and Periodic Cosine Operator Functions. Definition 1. A COF or SOF is said to be a.p. (uniformly a.p.) if for all x E E the corresponding function C(t,A)x or S(t,A)x is a.p. (uniformly a.p.). 5.2.1. [165] If E is weakly sequentially complete, then a weakly a.p. COF is almost periodic. T H E O R E M 1. [287] A COF C(t,A) is almost periodic if and only if the following three conditions are satisfied: (i) the COF C(t,A) is uniformly bounded; (ii) or(A) C R_; (iii) the set of linear combinations of the characteristic vectors of the generating operator A is dense in the space E. If, under these conditions, # E a(A) is an isolated spectral point, then # is a simple pole of the resolvent R(A.A) and

E=~(laI--A)~W(Izl--A).

5.2.2. [169] A COF C(t,A) is a.p. if and only if AE,7$-(.-t/2,0) and the system of characteristic vectors of the generating operator A is complete in the space E. T H E O R E M 2. [82] Cauchy problem (3.2.1) has an a.p. generalized solution for all u ~ u i E E if and only if conditions (i)-(iii) of Theorem 1 are satisfied and 0 E p(A). T H E O R E M 3. [82] A COF C(t,A) or an SOF S(t,A) is uniformly a.p. if and only if the following three conditions are satisfied: (i) the COF C(t,A) or SOF S(t,A) are uniformly bounded with respect to t E R; (ii) the set (1/i)a(A) is a harmonious subset in R and 0 E p(A). (iii) the set of linear combinations of characteristic vectors of the operator A is dense in E. 5.2.3. [82] If the COF C(t,A) is uniformly a.p., then or(A) consists of simple poles of the resolvent R(A,A). In this case a(A) = Pa(A). 5.2.4. [253] The following conditions are equivalent: (i) the COF C(t,A) is periodic as an operator function; (ii) the COF C(t,A) is strongly periodic; (iii) the COF C(t,A) is weakly periodic. T H E O R E M 4. [82, 218, 305]. The COF C(t,A) is periodic with period 27r if and only if the following three conditions are satisfied: (i) a(A) _C (l ; l = --k 2, k ~ Z); (ii) the spectrum or(A) consists of simple poles of the resolvent: (iii) the set of linear combinations of characteristic vectors of the operator A is dense in the space E.

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When conditions (i)-(iii) are satisfied the Riesz projectors are given by the formulas

1

p(__k2).X__-

cos(ks)C(s, A ) x d s for k ~ O , o

I

2-~- C(s, A)ds

for

k=0,

0

where, for x E D(A), oo

C (t, A ) x = Z c o s ( k t ) P ( _ k a ) x

'

(2.1)

k=O

where the series converges uniformly with respect to t ~ R. 5.2.5. [218] When E = H and C(t,A) is 2r periodic, Eq. (2.1) holds for all x E E, and convergence of the series is uniform with respect to t ~ R. T H E O R E M 5. [83] The function C(t,A)u ~ + S(t,A)u 1 is 27r periodic for all u ~ u 1 ~ E if and only if conditions (i)-(iii) of Theorem 4 are satisfied and 0 E p(A). 5.2.6. [218] A COF C(t,A) is periodic with period T if and only if the function F(z) := (1 -- e--TZ)zR(zZ,A) can be analytically continued to an entire function i:(z) such that for tzl > r we have

II~(z)ll~<Mexp(qlzl2-~),

wt~ere q, hi, r, e > 0 .

(2.2)

5.2.7. [218] A uniformly well-posed Cauchy problem (2.1) for n = 2 has only periodic solutions of period T if and only if A 6 ~ ( M , co) and the function F(z)/z (see Prop. 2.6) can be analytically continued to an entire function Q(z) such that for Izl > r bound (2.2) is valid for the function Q(z). 5.2.8. [215] Assume that a COF C(t,A) is given in a Hilbert space H and C(t,A) is weakly a.p. Then C(t,A) = Q-1C(t,V)Q, where V is a self-adjoint operator, V := P.~>0,XP(,X), and P(A) is a family of orthogonal projectors. 5.2.9. [305] If a COF C(t,A) is periodic for every x E D(A), then C(t,A) is periodic. 5.2.10. [7] Let (a(--A))U2 rq R+ be no larger than countable. Then all of the solutions of problem (3.2.1) will be almost periodic if and only if the following conditions are satisfied: (i) the COF C(t,A) is uniformly bounded for t E R; (ii) 0 E p(A); (iii) for every limit point Xo of the set (o(A))U2 there exists a sequence ~n ~ R that converges to zero and is such that lim en(~,--ri;~o)[(enq-i~o).I--A]-~x-~O - for every x ~ E. 5.2.11. [7] A COF C(t,A) is a.p. in the uniform operator topology if and only if it is uniformly bounded on R and (a(A)) 1/2 is a harmonious subset of R. 5.2.12. [7] Let A 6 ~ ( t ~ , 07) and assume that a(--A) has no limit points in R+. Then (i) the linear hull of characteristic and root vectors of A is dense in E if there exists a function X(t) such that

llC(t,A)ll~x(t) aria x ( t ) ~ k ( l q - l t l ) '

for t6R, ~I~0;

(ii) when condition lim X(t)/t = 0 is satisfied, the COF C(t,A) is periodic with period 1 if and only if a(A) C_ {--(2rk) 2, k E N). 5.2. Supplementary Comments. Proposition 1.21 is also true when A(t) : R --* B(E) with separable E (see [428]).

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6. C O M P A C T N E S S IN T H E THEORY OF OPERATOR S E M I G R O U P S AND C O S I N E O P E R A T O R F U N C T I O N S We will denote the set o f c o m p a c t operators defined on E by Bo(E) (or Bo(E,F) if we are dealing with different spaces). 6.1. C o m p a c t n e s s in the T h e o r y of O p e r a t o r Subgroups. Definition 1. An OS exp(tA) is said to be c o m p a c t beginning (or to b e c o m e c o m p a c t ) at t o if, for t > t 0, the operator exp(tA) is compact. If t o = 0 then the OS exp(tA) is said to be compact. 6.1.1. [126] Assume that an OS exp(tA) becomes compact. Then f o r any n u m b e r c~> 0 there exists a factorization of the space E = Ef 9 E d such that Ef and E d are invariant under the OS exp(tA), the subspace Ef is finite dimensional, and the restriction of the OS exp(tA)~, i to E a satisfies the condition

lim e~t exp ( t A IEcl) '

~

0

9

6.1.2. [126] If the generating operator o f an OS exp(tA) has a c o m p a c t resolvent, and the OS itself is continuous with respect to norm at the point to, then the OS exp(tA) is c o m p a c t beginning at t o.

THEOREM 1. [124] An OS exp(tA) is c o m p a c t if and only if it is continuous with respect to n o r m for any t > 0 and the generating operator has a c o m p a c t resolvent. 6.1.3. [124] If an OS exp(tA) is compact, then intersection or(A) n {z ~ C : a < Re z < b} for any fixed a, b E R contains a finite n u m b e r o f characteristic values o f the operator A. 6.1.4. [126] If an OS exp(tA) is compact, then the spectrum o(A) = Po(A). We define corn(A) := {t >0 : exp(tA) - - I is a compact operator }. 6.1.5. [237] if C o m ( A ) 4: ~, then the OS exp(tA) is invertible (t >_ 0). THEOREM 2. [237] For an OS exp(tA) the following conditions are equivalent: (i) C o m ( A ) = 1~+; (ii) the generating operator A is compact; (iii) the operator AR(~,A) - - 1 is compact for all A > w(A). 6.1.6. [237] If Corn(A) is a dense subset o f R+ without interior points, then A is u n b o u n d e d . 6.1.7. [237] If a generating operator A is u n b o u n d e d and Corn(A) 4: ~, then limx.+0+ ]I exp ( r A ) q I [[ > 2. Definition 2. An OS exp(tA) is said to be quasicompact if lim,_~:~ in: 1il exp ( t A ) - - K If: KEBo ( E ) = O. 6.1.8. [126] For an OS exp(tA) the following conditions are equivalent: (i) the OS exp(tA) is quasicompact; (ii) Ew(exp(tA)) < 0 for any t _> 0; (iii) there exist t o > 0 and K E Bo(E) such that II exp(toA) -- K II < 1 6.1.9. [126] Let exp(tA) be a quasicompact OS. Then the set {A E a(A) 9 Re A _> 0} is finite (possibly empty) and contains only poles of finite multiplicity. If we denote the poles by Az . . . . . we obtain exp (tA) = T~ (t) + . . .

Am

and their multiplicities by K(1) ..... k(m),

q- Tm (t) -}-Rm(t), t~O,

(1.i) :.(t)-I where

T,(t)=exp(~,,t)

~ V(A--~.,I)JP(~,)/jr. t>O and HRm(t)II -< c e - "

for some constants c and s > 0.

]=0

6.1.10. [126] Assume that the OS exp(tA) becomes compact. T h e n the spectrum or(A) is a countable set {A:,Az,...} (it may be finite or e m p t y ) and contains poles of only finite multiplicity. Moreover, the set {# E a(A) : Re # >_r] is finite for only r G R. If we denote the multiplicity o f the poles by k(l) and the corresponding projectors by P(Ak), we have, for each m, expansion (1.1), where IlRm(t)I[ -< c exp((s + Re Am)t), (Re Ap+ 1 < Re Ap) for some constants c and ~. 6.2. C o m p a c t n e s s in the T h e o r y of Cosine O p e r a t o r Functions.

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Definition 1. A C O F C(t,A) is said to be compact if~the operator C(t,A) is c o m p a c t for any t ~ R. An SOF S(t,A) is said to be c o m p a c t if the operator S(t,A) is compact for any t E R. 6.2.1. [397] If an operator C(t,A) is c o m p a c t for every t ~ (ct,fl) for some cz < fl, then the C O F C(t,A) is compact. 6.2.2. [397] If an operator S(t,A) is c o m p a c t for every t E (a,fl) for some a < fl, then the SOF S(t,A) is compact. 6.2.3. [307] If dim E = oo and A is u n b o u n d e d , then the operators C(t0,A ) and C(2to,A ) cannot simultaneously be compact for any t o > 0. 6.2.4. [397] U n d e r the assumptions o f 6.2.1, it is necessary that dim E < c~. T H E O R E M 1. [397] The following conditions are equivalent: i) the SOF S(t,A) is compact; (ii) the resolvent R(A2,A) is c o m p a c t for any A > we(A). 6.2.5. [397] Assume that an SOF S(t,A) is compact and the generating operator B E B(E). Then the operator A-t-BEg(M, o~) and the associated C O F is compact. T H E O R E M 2. [ 8 1 , 2 5 0 ] The following conditions are equivalent: (i) the generating operator A is compact; (ii) the operator A2R(A2,A) - - I is c o m p a c t for every A > we(A); (iii) the operator S(t,A) - - tI is c o m p a c t for any t ~ R; (iv) the operator C(t,A) -- I is c o m p a c t for any t E R. 6.2.6. [250] Assume we are given an operator C(t,A) -- I that is compact for an3: t ~ R. Then operator AR(A.A) -# R ( # , A ) is c o m p a c t for all A, # E p(A) Such that Re A, Re # > we(A). 6.2.7. [250] Let C(t,A) - - I be c o m p a c t for every t E (a,fl) for some c~ < ft. Then the operator C(t,A) --I is compact for any t E R. 6.2.8. [250] If S(t) - - tI is a c o m p a c t operator for every t ~ (c~,fl) for some a < fl, then the operator S(t) -- tl is compact for any t ~ R. 6.2.9. [810] If S(t,A) is an SOF and the o p e r a t o r C(t,A) -- I is compact, then the space E must be finite dimensional. 6.3. S u p p l e m e n t a r y R e m a r k s . T h e o r e m 6.1.i can be transferred to the case of nonlinear OS's [264]. The property Com(A) q: r is essential for the study of M a r k o v semigroups [237]. In [357] there are sufficient conditions for the operator

A= ~

(--1)iDJctj~D~+q

ljl,tkl~l

with q(x) ~ oo as x ~ oo to have a c o m p a c t resolvent. The case o f c o m p a c t R(A,A) -- R(Ao,A ) is also considered, but here it is the behavior o f the natural spectra with perturbations that is discussed. See also Propositions 2.2.5, 2.5.7, 2.6.7, 2.6.13, 2.7.46, 2.8.32, 3.1.35, 4.1.12, 8.1.4.; and 8.1.21. 7. U N I F O R M L Y B O U N D E D O P E R A T O R S E M I G R O U P S AND C O S I N E O P E R A T O R F U N C T I O N S For every element y ~ E we define the adjoint set 0(y) : ----{y*eE*

:[ly*ll2=llylF=
7.0.1. [301] The set 0(y) is n o n e m p t y for all y E E, convex, closed, and such that 0(0) = {0}. Consider a m a p p i n g ~o : E -~ E* such that ~9(y) ~ O(y) for all y E E. Such a m a p p i n g ~ is called the dual map. We fix some ~p and set [x,y] := (x,~o(y)) for all x,y ~ E. The function [x,y], which maps E x E into R or C, is called the semiscalar product. 7.0.2. [301] The semiscalar p r o d u c t has the following properties:

Ix+y, z]=[x, z] + [y, z], [~ x, y] =~[x, y], [x,x]=llxll 2, I[z,y]l~llxll.[lylt for any x,y,z~E, 1092

Definition 1. A n operator A E L(E) is said to be dissipative (with respect to the semiscalar p r o d u c t [x,y]) if Re[Ax, x]~<0

for all

xED(A).

(0.1)

If the _< sign is replaced by _> in (0.1) we say that A is accretive.

Definition 2. By the numerical domain o f an o p e r a t o r A we mean the set

W(A) : = { [ A x , x] :xED(A), Ilxll=l}. We also set O(A~:-- sup(Re A:A E W(A)). 7.0.3. [44] If an operator B E B(E) is dissipative, then the resolvent R(A,B) exists for any A such that Re ,~ > 0.

Definition 3. A dissipative operator A is said to be m-dissipative if p(A) r R+ q: ~. The operator - - A is said to be m - a c c r e t i v e if A is m-dissipative.

7.1. Uniformly Bounded Operator Semigroups. T H E O R E M 1. [301] An operator A generates a contraction OS if and only if D(A) =E and A is m-dissipative. 7.1.1. [301] An operator A is dissipative if and only if for any cz > 0 the operator I - - czA is injective and [1(I -O~A)-1 II -< 1 7.1.2, [126] Let A be a dissipative operator. Then: (i) if A is closed, then the closure A is also dissipative; (ii) if D (A)@~(A) , in particular, if A is densely defined, then the operator A is closed; (iii) A is m-dissipative if and only if ~(1--hA) =E for all h > 0.

Definition 1. A n operator is said to be T - a c c r e t i v e if it satisfies the condition

[l( (l--~.m ) - x u - (I--~,A )-~o)+It~II (u--o)+ll for A > 0 and u, v E E. 7.1.3. [160] A n y T - a c c r e t i v e operator in a Banach structure E is accretive if and only if dim E is less than three or the norm is p - a d d i t i v e for p _> 0, i.e., II x § y II p -- II x IIp § II y II ~ (p< o0) or II x § y II = max( 1[x II, II Y II) (p = ~ 7.1.4. [213] Assume that A 6 ~ ( I , 0 ) .Then

Re=O

for a l l

x6D(A) and yEO(x),

(1.1)

and the equation

(~..I--A)D(A) =E

(1.2)

is satisifed for all A E R (A :~ 0). 7.1.5. [213] If A is densely defined and both (1.1 .) and (1.2) are satisifed for Ao > 0 and A1 < 0, then A O f f ~ ( 1 , 0 ) . 7.1.6. [126] The contraction group l[ exp(tA)[1 _< 1 is an isometry, i.e., II exp(tA)x II = I1x It for all x E E, t E R. Definition 2. A n operator A is equivalent to an m - a c c r e t i v e operator if there exists an equivalent n o r m in which it is m-accretive. 7.1.7. [296] Let A and A 2 both be equivalent to m - a c c r e t i v e operators. T h e n A generates a u n i f o r m l y b o u n d e d analytic OS with angle 7r/4. In this case the operator A s generates an analytic OS with an angle O if and only if A generates a u n i f o r m l y b o u n d e d OS with angle 7r/4 + 0/2 (0 < O < ~r/2). 7.1.8. [296] Let A sk be equivalent to an m - a c c r e t i v e operator for bounded analytic OS with angle ~r(1 -- 2 - n ) / 2 .

k=0, n..

T h e n A generates a u n i f o r m l y

Definition 3. An operator A is said to be totally accretive if for every natural n ~ N the operator A n is accretive. Total m-accretiveness is analogously defined. 7.1.9. [296] The following conditions are equivalent: (i) A is equivalent to a totally m - a c c r e t i v e operator; (ii) A generates an analytic OS with angle 7r/2; 1093

(iii) for all n ~ N the operator A n generatives an analytic OS with angle 7r/2. 7.1.10. [296] In a Hilbert space H the following conditions are equivalent: (i) A is m-accretive and totally accretive; (ii) A is totally m-accretive; (iii) A is self-adjoint and positive. 7.1.11. [296] If A is m-accretive in a Hilbert space H, there exists only one m-accretive B such that B2 = A. If A has spectrum a(A) and W(A) belongs to ~2(0), then a(B) and W(B) are contained in ~(0/2). 7.1.12. [88] Assume that the operator semigroups exp(tA),exp(tB), exp(t(A+B))E~(1, 0) . Then

exp (t (A + B)) x = limk_~o {exp ( t

A)exp ( t

B)}kx

for each x E E and uniformly with respect to t in any compact set of R. 7.1.13. [229] Let AO$(M, 0), xt~ff(A), x2OY2(A). Then t

+ I exp(sA)(xl + x 2 ) ds = x , + 0 ( 1 / t )

for t-+ oo.

0

7.1.13. [229] Let A6,~ (M, 0) and assume that E is reflexive. Then E =./F(A)~99~(A), ./F(A)flg~(A) = {0} and the projector P(0) : e--,-X(A) along ~(A} has norm fi p(0)II -< M Moreover, any x E E can be represented in the form

x=PtO)x +t I--P(O ))xGX (A) *~( A ) and t

limt-~oo t -1 I exp(sA)xds = P (O)x. 0

7.1.14. [229] For A f ~ (M, 0 ) the following statements are equivalent: (i) E=dt~ ; t

(ii) limt-~o~t-11 exp (sA) xds

exists for every x ~ E;

o

(iii) lim~o+/~. (k, A ) x (iv) limx_~o+R(X, A)x

exists for every x E E; exists f or every xEgt(A).

7.1.15. [229] Assume that family of contractions (onto E) {~(t)}, t ~ 0 , has properties ~ ( 0 ) =1, 7P'(0)x exists for all x ~ c E and the closure A of the operator Y ' ( 0 ) ] ~5 generates a contraction OS. Then for each x E E we have exp (tA) x ~- limk~:. T (t/k) k x uniformly with respect to t in any compact set of 1~+. 7.1.16. [360] Assume there exists a 6 > 0 such that Hexp(tA) -- III < 2 for all t ~ (0,8). Then for every t E (0,5) and x E E we have t

k

lim t -1 1 exp(sA)xds-----lim k -1 X e x p ( j t A ) x t ~oo

1094

0

k..,.oo

j~O

if one of the limits exists. 7.1.17. [44] Assume A~, A2, Ax +

AaEg(1,

0) . Then

s-j 2I x0(-2 7.1.18. [341] Let

AEff(M,0).

Then

llAxII2~4M211xll.llAexll, x6D(A2). 7.1.19. [229] Assume that under the assumptions of 1.18 the space E = H is a Hilbert space. T h e n for each x D(A z) we have

IIAxlI~2" ItxIt. IIA2xll,

(1.3)

where equality occurs if and only A2x + ,~Ax + ,~2x = 0 and Re(A2x,x) = 0 for some I E R+. If A generates an isometry group, the constant 2 in (1.3) can be replaced by 1. This inequality (along with Prop. 1.18) belongs to a series of m o m e n t inequalities (see [59]). Definition 4. An OS exp(tA) is said to be weakly (strongly, uniformly) ( C , a ) - e r g o d i c at infinity if the operator

CH(t. cOx:=~t-~f(t--sp-lexp(sA)xds

existsforall

t>0;

0

max(0,w(A)) and if the limit

i e-~.tiiCH(t,~z)xlldt < o~

forallx~Eand)~>

0

(C, ~)-Iim exp (tA): =

lim

t~oo

CH(t, ~)

exists in the weak (strong, u n i f o r m ) operator topology.

This is the so-called Cesaro limit. Definition 5. An OS exp(t,A) is said to be weakly (strong, uniformly) Abel ergodic at infinity

A-lira exp (,tA): = lim k 3 exp

(tA) dt = lira k R (L,

,4)

exists in the c o r r e s p o n d i n g operator topology. 133' setting t --* 0+ instead of t ~ ee or ), --, ~ instead of ), ---, 0+, we obtain the definition o f ergodicity at zero. 7.1.20. [101] An OS exp(tA) is weakly Abel ergodic at zero (infinity) if and only if it is strongly Abel ergodic at zero (infinity). 7.1.21. [101] If an OS exp(tA) with c~ >_ 0 is weakly (C,c~)-ergodic at zero, it is strongly (C,c~)-erogodic at zero. 7.1.22. [101] Let exp(tA) be an OS that is strongly Abel ergodic at infinity and A-limexp(tA)x:-=-Px for any x ~_ E. Then (i) the operator P is a projector, i.e., P = p2; (ii) Pexp(tA) = exp(tA)P for all t > 0; (iii) PAx = 0 for all x E D(A) and APx = 0 for all x ~ E; (iv) Y ~ ( P ) = A ~ ; (v) E=~(P)~./F(P), ~ ( p ) n . h " ( p ) = { o } , E=Yt(A)O,/~'(A). 7.1.23. [101] The following conditions are equivalent for an OS exp(tA) with w(A) 5 0: (i) exp(tA) is strongly Abel ergodic at infinity; (ii) for each x ~ E the set {)~R(A,A)x}o<~,< w is relatively sequentially weakly compact; (iii) /~.R(k,A)!J~<M for 0 < ~ < c o and E=A~ For convenience we will denote the strong, weak, -weak, and u n i f o r m limits by s-lim, w - l i m , w*-lim, and ulim. respectively. 7.1.24. [101] The following conditions are equivalent for an OS exp(tA) with a~ _< 0: (i) the OS exp(tA) is u n i f o r m l y Abel ergodic at infinity and u-(A)-lim exp(tA)=p , where p2 = p ~ B(E); 1095

(ii) the point k = 0 is a simple pole of the resolvent R(k,A) and the residue of R(A,A) at zero is P~ (iii) s. limL2R(t., .4):=O

and

E--~(A)~,A~

(iv)

and

~ ( A 2) isclosed.

s-lim;~,~,().,A)=O

and the subspace is closed;

;'.~0+

7.1.25. [101] If w(A) _< 0 and the OS exp(tA) is u n i f o r m l y Abel ergodic at infinity, then

:~(A h) =5~(A) for

keN. 7.1.26. [300] Let E be a G r o t h e n d i e c k space with the Dunford--Pettis property. If the space E / E o is separable, there exist operators B, P E B(E), such that I - - P is a projector of finite rank, PB = BP, and exp(tA) = Pexp(tB), t _> 0. 7.2. U n i f o r m l y Bounded Cosine Operator Functions. 7.2.1. [232] There exist operators A E ~ ( M , ~o), such that for any b ~ R the operator A + b.l does not generate a u n i f o r m l y b o u n d e d COF. For all x E E and a E l~+ we define the operator

? i (sin Fax:= .'7" ~ t(at) /]-' C (2[. A~xdt, 0

which is bounded (see [348]) when

A E ~ ( M , 0) .

7.2.2. [348] Let 0 _< a_< b. T h e n

FaFox=F~FaX = 2 t F=xdu__ (b--a) Fax.

xEE.

0 b

7.2.3. [3481 Assume that for some 0 _< a _< b the equation

2i

Ftxdt -=(b--a) fF~x+F~xt

is satisfied for all

c/

x ~ E. Then the open interval ( - - b 2 , - - a 2) __ p(A). 7.2.4. [348] The following equations are satisfied for the operator Fa with a ~ R+: a

F ~ ( r t - - 1 ) n I (a-ty'-~F,dt,

n=2,3

.....

0

a

exp (itFa) = I + itF~ -- t~ i exp (it (a-- s)) F,ds. 0

Assume that a C O F C(t,A) is such that the operator Ct"q-l a

gaX:= 1imEa,~x:= lira a~O+

~ [XR(Z 2, A ) - - LP,, (;',L

a~O-- aOiO

A)]xdk

(2.1)

(where A = a + ir) is linear and continuous for all a ~ 0. 7.2.5. [348] T h e r e are examples of u n i f o r m l y bounded COF's for which the family {E,~) of (2.1) does not exist. 7.2.6. [405] Assume that for the C O F C(t,A) the family (2.1) is defined and E~ = E b for some 0 < a < b. then

( - - b 2, - - a 2) N (Ro (A) UPo (A) ) = ~ . 7.2.7. [405] Let Ea,,~x be b o u n d e d for a ~ [0,d] and c~ E [0,c~] with any d, ~ _> 0. ~ 4= 0. Then, for all x E E and a E [0,~], E a exists, the o p e r a t o r E a is bounded, and for all 0 < a _< b we have E a E b = E b E a = 7riEa, where for almost all

1096

a~Owehave

2 i sin (at) C ( t , A ) x d t iftheintegralconverges. Now, wedefine, forO<_a<_b,A:=(--b2,--a 2) E~x =.-~-----i-0

and E a := E b - - E a. 7.2.8. [405] U n d e r the assumptions of 2.7 with any two intervals A 1 and &2 we have

7.2.9. [405] Let [1Er

I] -< const for any r ___0 and ~o > 0, where E a = E b for some 0 _< a _< b. T h e n ( - - b 2 , - - a 2) c_

p(A). 7.2.10. [405] Assume that under the assumptions of 2.7 Eax for any x E E is continuous at a o _> 0. The

--ao 2

Ra(A). 7.2.11. [348] Let E be reflexive and strongly convex with G a t e a u x d i f f e r e n t i a b l e norm, A6~'(M, 0), and let the operator C(t,A) for any t E R have a real spectrum. Then Ra(A) = ~. 7.2.12. [12] Let A~gq(1,0) and let ~ be the operator of 3.2.1. Then, for t ~ [In 2,oo), '11C (g, A) :j.e(e,e) < 1.

i! C ' (t, A)I!,~m,,e~ ~ t. In 2,

I',S (t, A)~lB~e,e:~
and

Aft(M,

for

ReZ>ln2.

0) . T h e n

sup ,I S(*. A)il < t~R

@dist (0, 1

a( A ) ) s u p ~blC it, A)11. t~

7.2.14. [7] In order for the set ~:--{xEE: supHS(ts A)xll < ~ } to be dense in E, it is necessary and sufficient that one of the following conditions be satisfied: (i) lira efl;?(en, A ) x = O for any x ~ E and some sequence 8n ~ R+ such that lira e n = 0 " ; (ii) the set ~ ( A ) is dense in E; (iii) W ( A * ) = {0}.. 7.2.15. [169] An operator A 6 ~ ( M , 0) satisfies condition (F) if and only if the following condition is satisfied on every bounded interval [a,b] (see Definition 3.2.2): sup {ITexp (tG [zaG, ~1)lID(G,u) : rt6N,

tE[a,b]} < ~ .

7.2.16. [215] Let E = H. Then there exist a s e l f - a d j o i n t operator and constant M > 0 such that (el/2(2bl + 1)) -1 9I _< Q _< MI and the operator QC(t,A)Q -1 is s e l f - a d j o i n t for every t ~ R. In this case C(t,A) = Q - l c o s ( t G ) Q and G" = B _> O(G := Q A Q - 1 ) . 7.2.17. [236] Let A6W(M, 0). For any x=g-k-zEY~(A)@ W(A)

] T 03

C(t, A ) x d t - - z = O ( i T l - 1 ) ,

II+i S(t, A ) x d t - - TTz

=O(IT

[-*) ~s

T~ce.

0

1097

7.2.18. [236] When

.4E~(M, 0)

E=~(A)~.h~(A), , where,

and E is reflexive we have

for each x ~ E. we

T

T-I1C(t,A)xdr

have strong convergence of 7.2.19. [236] When

A@C(M, O) supi

as T ~ oo.

and x e D(A) we have the inequality

S(t. A) Ax,i2~<4 supli

t. 0

For x E E we set

C ( t , A) Ax,I supli C (t, A)x,I.

t.b0

xEE PcX: = l i r a t-~S(t,

t~0

A)x, n-I

P~x: =

lim X ~ e-ztC ~.~ O-r

(t. A)xat

and P~x: = l i r a n1

')0

tz~r

~" C (kt,

A)x.

k=0

7.2.20. [370] The two operator Pc and P~ coincide and are projectors. We have

A~(C(s, A)--I),

~(Pc)=Ae(A)=.f'I s>0

U $l(C(s, A)--I),

.#'(PJ=~(Aj=

s>O

D ( P c ) = 71A~

A)--I)eU

s>0

~(C(s,

A)--I)=

s>0

={xEE:a{tn}, tn-~ ~ ,

w-lim(S(t.,

A)x)/t, exists}.

7.2.21. [370] For ever), t > 0 the operator Pt is a projector and

~(p,)=se(c(t,

A)--Z),

(t, A)-- I ) ~ (C (t, A)-- 1)=

D (Pt)=A~ =

ae(P,)=~(C(t, A)--Z)

xEE:a{nk}, k ~ oo,

~o-ltmnEl

~ C (It, A)x

exists

9

7.2.22. [370] Assume that there exists a 6 > 0 such that the operator C(t,A) + I is invertible for t ~ (0.6) (in particular, this occurs when Ii C(t,A) -- Ill < 2 ) The Pt = Pc for all t E (0,26). 7.2.23. [306] Let A e ~ ' ( l , 0 ) , and

A'

Then

lira

Cm(t)x~-C(t, A)x

for all x E E a n d t[ [ C m ( t ) - - C ( t , A ) l x

It -< t2 It Ax []/m 1/2 for all x E D ( A ) ,

m > 2.

m--t.~

7.2.24. [327] The functions C(t,A) and S(t,A) are uniformly bounded for all x E E if and only if there exists a constant M ~ 1 with the property

dk

(k = o , 1, 2 . . . . ).

1098

d k_t

Mk I

7.3. Supplementary Comments. See also Propositions 2.2.12-13, 2.2.16, 2.7.22, 2.7.29, 2.7.46-48, 2.7.52, 4.2.5, 5.1.7, 5.1.8, 5.1.14, 7.1.4, 8.1.14. Formula (1.20) was generalized in [257]. In [59,418] there are different forms of the moment inequalities for generating operators of OS's and COF's (see Proposition 7.1.18), and, for certain cases, the coefficients appearing in the inequalities are presented in explicit form.

8. THEORY OF PERTURBATIONS FOR OPERATOR SEMIGROUPS AND COSINE OPERATOR FUNCTIONS If an operator

A~'g (E), then by a perturbation of this operator we mean a linear operator G for which D(G) -__a

D(A). The operator A + G is said to be perturbed. If B E B(E), then it is clear that Aq-BE~ (E) 8.0.1. If a perturbation G is a closed operator, then there exist constants a and b such that

[IGx[l<~allxH+bliAxll

for any

xED(A).

Definition 1. If the preceding inequality is satisfied for some perturbation G, we say that the operator G is bounded with respect to A (or simply A-bounded, or subordinate to A). The smallest value b 0 of all possible constants in the preceding inequality is called the A - b o u n d of the operator G (or the norm of G relative to A). 8.0.2. [388] A-boundedness of an operator G is equivalent to D(G) _DD(A) and GR(A,A) E B(E) for some ), E p(A). 8.0.3. [44] Let A, G E L(E), where G is A - b o u n d e d with an A - b o u n d less than 1. The operator A + G can be closed if and only if the operator A can be closed. In this case the closures of the operators A and A + G have the same domains. In particular, A + G is closed if and only if A is closed. Let ,?,(~) denote either an OS or a COF. Definition 2. A class of operators ~t(~) is said to be a class of Miyadera perturbations for the family ~(t) with generator A if for any operator

GE~(~)

we have D(G) _~ D(A) and there exists a constant K > 0 such that

!

ii O| ~o WeuseKo~todenotethelimitof

K>.:=sup

;dt~
xED(.'t).

fox" any

e-Z O~(t)xldt:iixli=l, xED(A),

because Ka monotonically decreases with A and is positive. Definition 3. An operator GE'~ (E) is said to be subordinate to the operator i f D ( G ) _ D ( A ) a n d !IGxH -
asA~;thislimitexists

A e~ (E) with order a (a E [0,1])

is said to be totally subordinate to an operator

AGoG(E), if

it

is subordinate with order 1 and for any sufficiently small r/> 0 we have

[:G.v'~[~Jn(x)+qilAx!i

s

any

xED(A),

where <1% is a continuous convex functional on E. 8.1. Theory of Perturbations for Operator Semigroups. THEOREM 1. [44]. Let AE,~'(M, ~o) and B ~ B(E). Then A+BE~'(M, o+M[IB[[) . Also, the OS exp(t(A + B)) with fixed t > 0 is a holomorphic function of B. Jn particular, exp(t(A + zB)) is an entire function of the complex variable z. 8.1.1. [213] Theorem 1 is also valid for groups of operators. 8.1.2. [193] The perturbed OS in Theorem 1 satisfies the equation

e~

exp (t (A -r- B)) = exp (tA) '-- t exp ((t - -

s)A)B exp (s(A + B))ds,

t~0

0

1099

and can be represented as a series that converges absolutely and uniformly with respect to t 6 [0,T],

exp(t(A+B))=

x ' S~(t), rae S o ( t ) - - e x p ( t A ) , n~O

or n ( t ) = i exp (( t

-

-

(1.1) s)A)BS,~_I(s)ds,

tzEN,

0

or, in a different form (see [193]), exp (t (A + B ) ) = exp ( t A ) - -

.

.

.

.

.

(J .2)

.

9.. exp((t--h)A)dG+~ . . . dt 1. 8.1.3. [341] Let

A6ff(M, to) and B e B(E). Then

Ilexp(t(A.B))--exp(tA)ll~Mexp(oJt)(exp(M.]!Bi!.t)--l),

t~O.

8.1.4. [34, 126] Let exp(tA) be an OS that is either analy'tic, or continuous with respect to norm. or compact, and B E B. Then the OS exp(t(A + B)) is, correspondingly', either analytic, or continuous with respect to norm, or compact. 8.1.5. [126] Proposition 1.4 remains valid if B e B ( Y ) ( A ) ) . If in this case, we also have -BEBo(Y)(A)) and the OS exp(tA) is continuous with respect to norm, then the OS exp(t(A + B)) is also continuous with respect to norm. 8.1.6. [44] Let AG~ (M, co) , and assume that the operator G is bounded relative to A. Then, generally speaking. the operator A + G does not necessarily generate an OS. 8.1.7. [44] Let A, G6gq(1, 0) and assume that operator G is A - b o u n d e d with A - b o u n d b o < 1. Then

A-,-GE

~(1,0). 8.1.8. [2291 Assume that A is m-dissipative and densely defined, and that G is m-dissipative and A-bounded with A - b o u n d b o = 1, while G" is densely defined. The A + G is m-dissipative. 8.1.9. [229] In Proposition 1.8 the condition of denseness of D ( G ' ) in E" can be replaced by reflexivity of the space E. 8 . 1 . 1 0 . [44] Assume that A, G65~(1, 0) , the set D(A) n D(G) is dense in E, and the operator A + G + A- 1 has a dense domain for sufficiently large A. If we take the closure of A + G we have

A + G G ~ ( 1 , 0).

t

8.1.11. [35] Let

Afi~(M, to), U6~(E),

D(A)ZD(G)

, and

I!iOexp(tA)l!,'dt
where liG[] ':=

0

supCGx II/II x 11 : II x II -< 1, x ~ D ( A ) } Then the operator a + G has a closure and generates on OS that is representable in the form (1.1). 8.1.12. [316] If exp(tA) is a group of operators, and the operator G is such that D(D) = D(A), G is A-bounded 1

and

~lJOexp(tA)lt~dt<

do,

then ItGII,'is finite.

0

1

8.1.13. [193] Let AE~(M, to), D(G)~Eo

, and I ]] 13'exp(. tA)ii d t <

ce. Then D(A) C D(G) and A + G generates

0

an OS. 8.1.14. [229] Let A6ff(l, 0), assume that G is dissipative and totally subordinate to A. Then A + G 6 f f ( 1 , 0). 8.1.15. [316] Let A6ff (M, to) and Ge~l (exp (tA)), where GR(A,A) ~ B(E) for some A > ~o(A). Then there exists a finite or infinite % such that for each kl < % the operator A + , G generates an OS. The proposition remains valid for groups of operators.

1100

We now turn our attention to the fact that perturbations of the class

!IR may be u n b o u n d e d operators (cf. Prop.

1.12). 8.1.16. [10] Assume that we are given two w - c o n t i n u o u s OS's or C o - s e m i g r o u p s such that

Ilexp(tA)-exp(tO)ll=,o(t)

as

t--,-0.

Then exp(tA) = exp(tG) for all t >_ 0. 8.1.17. [10] G i v e n two w - c o n t i n u o u s groups or C o -groups exp(tAP) and exp(tG), the following conditions are equivalent: (i) there exist q > 0 and 61 > 0 such that

[]exp(tA)exp(--tG)--lll~l--e, (ii) there exist

g2, 62 > 0 and

for

0~t~6~.

P, W 9 B(E) such that W- 1 9 B(E), A = W(G + P)W - 1 and [1e x p ( t A ) W - l e x p ( - - t A ) W --

I11 < 1 - - E 2 f o r a l l t r

IV=b~'l exp(tA)exp(--tO)dt 6~

If these conditions are satisfied, then

and, consequently, [lI -- W II -< 1 - q .

0

Moreover, ]lexp (tA) W-'exp

(--tA) •-III

= ][exp (tA) exp

(-tO)-ItH-O(t)

and

[lexp(tA)Wexp(--tG)--Wtl=O(t) 8.1.18. [423] If the operators

as t---~0.

A, O e 5 (M, co) and the element .~ 9 D(A) are such that exp(sA)~ ~ D(G) for 0 <

1

s < t and

S HOexp(sA)x!lds<

~,

then

0 t

exp ( t O ) x - - e x p (tA) .) = i exp ( ( t - - s) O ) ( O - - A) exp

(sA) xds.

0

8.1.19. [126] There are examples of differentiable compact OS's exp(tA) such that for some B 9 B(E), exp(t(A + B)) is not continuous with respect to norm. 8.1.20. [126] Let exp(tA) be an OS that is continuous with respect to norm beginning with some t o and B 9 Bo(E), i.e., is compact. Then the OS exp(t(A + B)) is continuous with respect to norm beginning with t o . Definition 1. An OS exp(tA) is said to be quasicompact if lira inf {~;exp ( t A ) - - K ]J: KEBo( E)} -~ O. 8.1.21. [126] Assume that exp(tA) is a quasicompact OS and B 9 Bo(E). Then exp(t(A + B)) is a quasicompact OS. 8.1.22. [126] Let AE,~ (M, co) and

,B~B (.~)(A)).

Then the operator A + B generates and OS and, what is more,

there exists a B 1 E B(E) such that A + B is subordinate to A + B 1. More precisely, for some ,/o E p(A)

A+B=U(A+B~)U-L

~'here

U= (I--BR(Xo, A)).

1101

AEg~.(O,o~) and r ~ (03) there exist positive constants "7 and 6 such that if the

T H E O R E M 2. [44] For any

operator G is b o u n d e d relative to A with a, b < & then A + GEa~(0--e, ~/). a c ~ (0--~, 0) 8.1.23. [59] If AEa~(0, o ) , generates an analytic OS.

In particular, if 02 = 0 and a = 0. then

and the operator G is totally subordinate to A, then the operator A + G also

8.1.24. [341] U n d e r the assumptions of T h e o r e m 2 we have

Ilexp(t(A-~G))ll<~Ntexp((o+v(b)t), where

t)O,

lira v ( b ) = 0 . b~0

Definition 2. Let

AE~'(E).

A n operator K with D(K) R D(A) is c o m p a c t relative to A if there exists a Ao

p(A) such that the operator K.R(Ao,A) is c o m p a c t in E. 8.1.25. [11] Let

AEg~(0, co) and assume that the operator B is compact relative to A. If one of the sets D(A')

or D(B') is dense in E*, the A + B generates an analytic OS. 8.1.26. [11] Let A6a~ (0, o)) , and assume that the operator B is such that D(B) __ D(A) and R(Ao,A)B E Bo(E) for some Ao. T h e n the operator A + B generates an analytic OS. ]

8.1.27. [101] Let

AEa~(0,

and assume the operator G is A - b o u n d e d and

i ii G e x p ( t A ! i l d t ~ oo.

Then

0

A + G generates an analytic OS and (1.1) holds. 8.1.28. [219] Let A, GE~(M, o)) and Iim il exp

(tA)

_

If exp(tA) is a s e m i - F r e d h o l m OS.

exp ( t O ) i = 0

t ~0-+-

then: (i) the OS exp(tG) is s e m i - F r e d h o l m ; (ii) nul(exp(tA)) = nul(exp(tG)) = 0 if nul(exp(toA) ) = 0 for some to, nul(exp(tA)) = nul(exp(tG)) = oe if nul(exp(toA)) = oo for some to; (iii) def(exp(tA)) = def(exp(tG)) = 0 if dep(exp(toA)) < ~ for some t o, def(exp(tA)) = def(exp(tG)) = ~ if dep(exp(toA)) = ~ for some t o. 8.1.29. [59] Let A 6 ~ ( E ) and assume the resolvent R(),,A) has bound ii R(),,A)i~t _< M(1 + Lkl-~) for some 0 < fl < 1 and all A E p(A). If operator G is subordinate to A with order c~ ~ (0,13), then where

IIGR(A,A)II

IIR 0., A+G)

[1~ ~

(1 + I).l )-8 ,

-~ 8 for Re A ___02. 1

8.1.30. [219] Let exp(tA) be a s e m i - F r e d h o l m OS and let Then

A+.o~

L~ be A - s u b o r d i n a t e and f L~'"~ exp ( t A ) i J , d t < ~z. o

generates a s e m i - F r e d h o l m OS and Proposition 1.28 holds with

G=A+.~.

8.1.31. [219] Let exp(tA) be a s e m i - F r e d h o l m OS, and let G6~t (exp (tA)) and be A - b o u n d e d . Then there exists an E0 > 0 such that for kt < ~o the operator A + sG generates a s e m i - F r e d h o l m OS, where nul (exp (tA)) = nul (exp (t (A + e G) ) ) and

def(exp(tA))=dei(exp(t(A+eG)))

for a l l

t~>0.

8.1.32. [219] Let E be a Hilbert space, and assume AE.~(1, 0) , and the operator G is dissipative and Abounded with A - b o u n d b < 1, while inf {Re((A-t-O)x, x)}.;> - - oo. Then the OS's exp(tA) and exp(t(A + B)) have [x

~I

bounded inverses and def(exp(tA)) = def(exp(t(A + G)) for t >_ 0.

1102

8.1.33. [219] There are examples of F r e d h o l m OS's with an A - b o u n d e d operator G with A - b o u n d less than 1 and such that the OS exp(t(A + G)) is not Fredholm. 8.1.34. [101] Let exp(tA) be a positive OS in a Banach structure E, and let G be an A - b o u n d e d operator with 1

l i!Gexp(tA) '~dt <

o~. If, for some ~ E R, the operator G + ~I is positive, then the OS exp(t(A + B)) is also positive.

8.1.35. [178] Let exp(tA) be a positive OS, and let B ~ B(E) be a positive operator. T h e OS exp(t(A + B)) is irreducible if and only if J = (0} and J = E are closed ideals with the properties (i)

B(E,E|

exp(tA)J~J, (ii) BJ~J.

t~0;

8.1.36. [178] Assume that AEff(M,o~) and the space E is | with respect to A (see w Then there exists an OS U(t) such that for all t _> 0 and every x ~ E we have

Let B E

t

U(/)=exp

(tA) x+ f exp ((t --s)A)~*BU (s) xds. 0

Definition 3. Let B(E,F) be a space of linear bounded operators m a p p i n g E into F. Also, let [] U(t)l[ -< Mexp(~t), where~:=w+MI[BH.

8.1.37. [178] U n d e r the assumptions of 1.36, and with the same notation, U(t) = exp(tAX). T h e n E is O - r e f l e x i v e with respect to A x and

(i) D(A:')={xED(A- ):A - x , BxEE}, A•162 xeD (A ); (ii) D(A'*)=D(A*), A• x~ED(A• (iii) D(A'~)={xED (A**):A*x~-kB*xSEE}, A• -k-B'x:, x~-ED(AXS); (iv) D((A~):*)=D(A~*), (A*)~*x=A~*x+Bx, xED ((A~):*).

We write

& (t): := 1 exp ((t--s) o

A):*B exp (s(A+B)) ds,

K(k): =R(Z;

AZ*)B.

8.1.38. [178] Assume that U(t) is an operator function that is continuous with respect to norm on [to,OO) and K(A) is a compact operator function for sufficiently large k. Then the operator lJ(t) is c o m p a c t for all t > t o. 8.1.39. [178] In the notation of 1.38, if the operator function lJ(t) becomes compact, then Ea(A + B) < w(A). 8.1.40. [178] Assume E is | B E B(E,E| and for any k with Re k > S(A) the o p e r a t o r R ( A , A ~ is compact. Then the set {A E cr(A + B) : Re A > S(A)) contains no more than a countable n u m b e r of points (kk), each of which lies outside Ec~(A + B) or, equivalently, Ec,(A + B) c_ (,~ r C : Re ~ _< S(A)}. 8.1.41. [174] Assume that exp(tA) | and B r B(E'C.E ") are OS's on E e. Then the equation

exp (~ .-1) x ~ ~- exp (t A ) : x : @ t exp (( t -- s) A)*B exp (s~t)

x~ds, x:eE ~,

0

uniquelydeterminestheadjointoftheOSexp(t.~)onE'-9.

Here w*-lim / - ~ ( e x p ( t . 7 - 1 ) ~ l ) x a = A * L ~ z Z + B x ~ i f a n d o n l y

if x ~ -~ D(A'). 9 8.1.42. [ 174] U n d e r assumptions of 1.41, the operator 5X is part of operator A" + B in E @" (1.e., D ([4), .~ ( ) ) ~ E ~ ).

1103

8.1.43. [574] Let exp(tA) and exp(tG) be two OS's on E" that are w*-continuous for each t >_ 0. If t - % x p ( t A ) -exp(tG) tl = O ( t ) for t ~ 0+, then there exists an operator B ~ B ( E ' ) such that t t~

0 0

tk

,0

. . . B exp ((t~ - - t~) G) B exp ((t -- t~) G)dt~<... d tl, where the integrals are taken in the strong sense. Definition 4. We define the L ~ p e r t u r b a t i o n of an operator A by means of the following conditions: (i) let E and F be Banach spaces, and let D(A) be dense in E, where D(A), e q u i p p e d with the norm Il is complete (we will denote this space by (D(A),].[); (ii) A E B((D(KA),H),E) and L c B((D(A),H),F); (iii) ~ t ( L ) = F ; (iv) the restriction of A to W ( L ) (we denote it by A o) generates the OS exp(tA o) on E; (v) for a b o u n d e d o p e r a t o r 9 ~ B(E,F) we define the operator A~ thus: A~x := Ax, D(A~) := {x E D(A) : Lx = 69x}. 8.1.44. [242] Assume that there exist constants "/> 0 and Ao E R such that

IILxtl>~.~llx[I for a n

x6W(~.I--A) and ~ , > l o

Then the L q , - p e r t u r b a t i o n AO generates an OS for any cI, ~ B(E,F), ~5 ~ B(E,F). 8.1.45. [242] If inequality (1.3) in Prop. 1.44 is replaced by flLxl[ >- L~l.,~l[x [] for all

(~.31

xEWO..1--A)

and

A ~ {z 6 C : kl >__r, hrg zl < 1r/2} for some r, then AoEg{, (0, o~) implies that A generates an analytic OS. 8.1.46. [242] Let Aofio~(0, co). If the operator q5 in the Le; perturbation is c o m p a c t , then A~ generates an analytic OS. 8.2. T h e o r y of P e r t u r b a t i o n s f o r Cosine O p e r a t o r Functions. 8.2.1. [328, 413] Let AE~(/%l, co) and B ~ B(E). The operator A + B generates a C O F and ]!C(t,A + B) -C(t,A) II -" 0 as II B II --' 0 u n i f o r m l y with respect to any c o m p a c t subset of R. 8.2.2. [327] If, under the conditions of 2.1,

~ > coq- ~-M 1!B ii, there exists a n u m b e r s = t~l(w) such that

[] C ( t, A -? B) i~< ~l exp ( i t f ~ ), 8.2.3. [208] Let

tER.

A 6 ~ ( M , co). Then, for x E E, f

C(t, "2I - ~, A ) ~ = C ( t , A ) x 4 - ; t )

l'(;l"t"--s~) C(s, A ) x d s , 0

where 11 is a Bessel function and

iiC(t, A-k;-~I)ij--<Mch ( l - ~ t )

for -~EC.

Other more accurate estimates for the expression C(t,A _+~2I) are given in [208], 382-383] for Banach and Hilbert spaces. 8.2.4. [413] Let A E ~ ( M , ~o) and B ~ B(E1,E). Then A-k-BE~(M~, ~1) 8.2.5. [365,414] Let A E ~ ( M , co) and D(A) C_ D(G). If there exists w" ~_0 and M' _> 1 such that p(A) contains the set {z : z > w'} and the f u n c t i o n GR(z~,A) is infinitely differentiable, where 1

for x E E and any n E N and z > ~o', then A + G generates a COF. 1104

8.2.6. [229,232] If A, G ~ ( M , co)., then the operator A + G (or its closure) generally does not generate a COF, even when C(t,A) and C(t,G) commute. However, we always have A+BEa~(rt/2, co).. 8.2.7. [28] Let A, GE~(M, co) and assume that D 1 := D(A) n D(G) is dense in E. Then a "generalized" COF (in the sense that conditions (i)-(ii) of 1.4.1 are satisfied) is defined on D as follows: 1 t2

c Ct, A +G)x=C (t, A ) x + - U I ],(t Vl--s2, A) C (t.s, 9)xds, 0

I

where

j,(t,

A)x:=-~-

l,:l--s~C(ts, A)xds, xfiD,.

If A + G generates the COF C(t,A + G), then C(t,A + G) =

AO~(M, co) and the operator

Gem'(E) be such that

0

C(t,A + G), t c R. 8.2.8. [399] Let

(i) for the SOF, R(S(t,A)) _ D(G) for all t 6 R; (ii) the function GS(t,A) is strongly continuous with respect to t ~ R. Then the operator A + G generates a COF. In this case ~o

oo

C (t, A + O ) x = x Ck(t)

n

S(t, A+G)x=~_,S~(t),

k=O

(2.1)

k----O

t

where (~o(t): = C ( / ,

A),

t

C,(t): = I C(t--s, A)(}Sk_,(s)xds ,and .~o(t):=S (t, A), ~k(/):=.I S(t--s, A)O~,_,x 0

(s)xds,

0

and series (2.1) converges absolutely in B(E).

8.2.9. [399] Under the conditions of 2.8, oo

R(~.~,A-t- G)=R(X~, A)X (OR(~' A))k. k=O

8.2.10. [397] If, under the assumptions of 2.8, the SOF S(t,A) is compact, then the SOF S(t,A + G) is also compact. 8.2.11. [387] Let At, ,zI2E~'(M, co) and D(A1) _c D(A2). Then

t

C (t, At)x--C (t, A:)x= l S(t--s, Az)(Al --A~)S(s, AOxds 0

for all x E D(A1). THEOREM 1. [397, 399] Let A e ~ ( M , co) and G~'(E) (i) D(A) c D(G); (ii) there exists a continuous function D(t) such that

IIGS(t, A)xU~K(t)llxll

be such that

for a11 x~D(A).

Then the operator A + G generates a COF. 8.2.12. [399] Assume that the operator A~i~(M, co) satisfies condition (F) with the operator GE~(E) , and for some QE~(E) the condition D(G) c__D(Q) is satisfied. Then A + Q generates a COF. 8.2.13. [399] Assume that in Prop. 2.12 the condition of inclusion of domains of definition is replaced by the condition D(A) c_ D(Q). If the operator Q is G-bounded, then the operator A + Q generates a COF. 1105

G ~ ( C ( t . A)) . GR(A,A) ~ B(E), and kl< Koo-1. Then the operator A + G generates a COF, lira li C(t, .4-+-aG)--C (t, A)1] = 0 u n i f o r m l y on any c o m p a c t set in R, and 8.2.14. [374, 388] Let A c ~ ( M , c0).

C (t, A + e,Q) -- ~ ekff~(t), k=O t

where Co(t): ~ C (t, A);

C k (t) x = l C~_1 (t -- s) GS( s, A) xds,

xED (A), and Ck(t) is a continuous extension of Ck(t)

0

to all of E. 8.2.15. [3081 Let C(t,A1) and C(t,A~.) be COF's such that li C(t,Ax)II -< Me~t and I] C(t,A2)II -< Ne~t for t _>0 and [I(A1 -- n2)x 1[ -< a II x H + b 1]AlX 1] for x E D(A1). Then for z > co we have

il (c (t, & ) - c (t, A2))(z~Z-- A)-~ !i--< 9 MN J-~Qt sh(cot),

0)~V,

MN

"~ [ o~v.~ Q(ch((ot)--ch(vt)),

, .

o~,

where Q := (1 + M)b + (a +bw2)M/(z 2 -- w2). 8.2.16. [308] If a, b ~ 0 under the assumptions of 2.15, then

limC(t, A2)x~-C(t a,b~O

,A1)x for x~D(A1).

8.2.17. [371] Let K ( t , A ) be a w % c o n t i n u o u s C O F (see w and let B be a w % w ' - c o n t i n u o u s operator on E'. Then A + B is the w*-generating operator of a w*-continuous C O F K ( t , A + B) and lira HK(t, A+B)--K(t, A)[I = 0 uniformly on any c o m p a c t [0,T]. 8.3. S u p p l e m e n t a r y C o m m e n t s . Problems on perturbations of adjoint OS's were discussed in [174-177]. In was proved in [261] that when a ~ (0,1) the condition t - a ]l exp(tA) -- exp(tg)II = O(1) as t --, 0 is not sufficient for validity of Prop. 1.43, even for unitary OS's or c o m m u t i n g OS's, For i n f o r m a t i o n on limit values of perturbations in theorems on perturbations of F r e d h o l m operators, see [362]. n

Perturbation of the well- posed C a u c h y p r o b l e m

X Aju (t)(n-~)~O, t >~ 0, u(J)(0) ~ ]=o

u J, j = 0, n - - 1 was studied

in [132] by replacement of the operators A n by A~ + B(t), where B(t) : 1~+ --, B(E). This paper also considers restrictions on B(T) under which the p e r t u r b e d p r o b l e m remains well-posed. Concerning perturbations of the f o r m A 9 B, see [242] In [414] there are conditions, for the case E = C([a,b]), under which the assumptions of Prop. 2.5 are satisfied by operators Au: -~a(x)u" + ~1a , (x)u', D(A)={u:u, u'EC([[t, hi), u ' ( x ) = 0 with x =/~ and x = b) and Gu: ~-

( b ( x ) - - ~1 a' (X)) u' + c ( x ) u ,

D(O)={u:u,

u'EC([[z, b]):

u'EC([a,b])}.See

alsoProps. 2.1.34,2.1.7-9,2.1.17.

6.2.5, 7.1.12, and 9.2.7. 9. A D J O I N T O P E R A T O R S E M I G R O U P S AND C O S I N E O P E R A T O R F U N C T I O N S 9.1. Adjoint OS's. Let E* be the adjoint of a Banach space E. For each m e m b e r x ~ E the f o r m u l a (x,y ~) defines on the space E* a functional x** ~ E** by means of the rule (y",x "~) := (x,y*). We call the m a p p i n g ~r : x ~ x "~ the natural mapping of E into E**. 9.1.1. The natural m a p p i n g ~r is isometric, i.e., ![ x l[ = 1[x'~ ]l for all x E E. Assume that the operators of exp(ta) ~ are adjoint to the operators of the OS exp(tA). The family exp(tA)" satisfies conditions (i)-(ii) of Def. 1.3.1 and the bound ]i exp(ta)" II < Me~t, although it generally does not necessarily

1106

satisfy condition (iii), i.e., it is not necessarily a C o - s e m i g r o u p in E" (continuity is taken to be with respect to the strong topology in E'). 9.1.2. In the case of a reflexive E, the assumption

AEff(M, co) implies that D(A*) is dense in E*. This is not

so in the general case. 9.1.3. [101l (R))~,A*) = R(,~,A)*. Let E e denote the (closed) subspace of E" on which the f a m i l y exp(tA)* is strongly continuous with respect to t, and define the OS exp(tA) e to be e x p ( t A ) * ]e~. 9.1.4. [174,424] for an OS exp(tA) we have (i) the m a p p i n g t --* exp(tA)~x * is w*-continuous with respect to t ~ 1~+ f o r all x* c E*; (ii) the m a p p i n g in (i) is w ~ - d i f f e r e n t i a b l e f r o m the right at zero if and only if the element x* E D(A*); (iii) if x* ~ D(A*), then if t > 0 we have exp(tA)*x* E D(A*) and the f u n c t i o n in (i) is w * - d i f f e r e n t i a b l e , where

d

a---i < X, e x p ( t A ) * x * > = < x, A* e x p ( t A ) * x * > --= < x, e x p ( t A ) * A * x * > for all x E E a n d t >_0; (iv) the domain D(A | is w ' - d e n s e in E*, where A | is the generating operator of an OS exp(tA)* defined on Ee; (v) the family exp(tA) ~ operates according to the f o r m u l a exp(tA)* = G" I - - A*) exp( t A) | (~. I - - A*)-'. 9.1.5. [178] The following statements are true: (i) E e is the closure with respect to norm of D(A'); (ii) E(A e) = {~ ~ D(A ~) : A ' ~ E Ee}; (iii) A e and A* coincide on D(AO). T H E O R E M 1. [4241 For AEN(M, ~o) we have: (i) the o p e r a t o r A | is the largest restriction of A ~ with domain and range in the subspace Ee; (ii) the subspace E ~ = D ( A * ) , where the closure is taken in the strong topology of E*; (iii) the subspace E e is invariant relative to the family exp(tA) ", and the a b o v e - d e f i n e d f a m i l y exp(tA) e is a C osemigroup of operators on E~ (iv) if E is reflexive, then E e = E" and A e = A ~. We now introduce a new norm in E: [i x [l': sup { i < x, x ~ > l: x~EE |

J] x'~ l] < 1} for x ~ E .

9.1.6. [300] Let E be a G r o t h e n d i e c k space. Then the families exp(tA)* and exp(tA) *~, t ~ 1~+, are strongly continuous in E ~ and E "*, respectively. 9.1.7. [178] Let ~:=limliZ,/?(z, 4):,i. Then

iixii'~.~llx!i~.~Citxl]'

for any ' x E E .

These norms coincide for contraction semigroups. 9.1.8. [158] Let sector 2(0), and

A6a~(0. o)).

Then the operator function exp(tA)" has a h o l o m o r p h i c continuation into the

for any

(i) e x p ( ; , A ) * e x p ( r (ii) w*-ltmexp(~A)*x*-~-x*

D e f i n i t i o n 1. A Banach space E is said to be |

for a l l

r r

X*EE* and ] a r g ; I - . < ~ < 0 .

(solar reflexive) according to AE~(M, o~), if E = E e e . 1107

9.1.9. [175] A space E is Q-reflexive with respect to an operator AEff (M, co) if and only if one of the following conditions is satisfied: (i) the resolvent R(~,A) is a ( E , E ~ compact for all ;~ E p(A) (here a(X,Y) is the locally convex topology on X induced by the functionals of Y); (ii) the space E ~ is Q-reflexive with respect to the operator A. Definition 2. We will call a subgroup in E* that is continuous at zero in the w*-topology a Co*-semigroup. 9.1.10. [10] For a Co*-semigroup exp(tA) there exist constants M _> 1 and 13 _>inft>o(t-Xln [] exp(tA)II) such that

Ilexp(tA) II~M exp(13t), t~O. 9.1.11. [101 For a C o -semigroup exp(tA), assume we are given a set ~)~D(A), that is w*-dense in E* and invariant under exp(tA). Then the ! ~ -core of the operator A, i.e., the w* -- w* -closure of the restriction A lg ) , coincides with A. Definition 3. We say that ~ E E is holomorphic for an operator A if 2 E D(A ~176and the function

co

z ~ ~ zk [IAkxll/kt, zGC k=0

is analytic. * * 9.1.12. [10] An operator A6~'(E*) is the generator of a C o -group if and only if it is w*-densely defined, w -w* closed, and satisfies the following conditions: (i) there exist M >_ I and fl > 0 such that

II(l--czA)~xll~M-'(1--r

for a n x6D (Ak), a l l a with

1~113<1

and k(~N;

(ii) either I~(I--aA) for all c~ with 0 < c~fl<1 coincides with the space, or A has a w*-dense set of holomorphic elements and any m e m b e r of the space can be approximated by a uniformly bounded net of holomorphic elements. 9.1.3. [10] For a Co*-semigroup exp(tA) the following conditions are equivalent: (i) x E D(A); (ii) sup{ tl (exp(tA) --I)x II/t : t c [0,1]} < oo. The analog of this statement for OS's is generally false. 9.1.14. [10] For a C o- or Co*-group exp(tA) the following conditions are equivalent: (i) there exist e > 0 and 6 > 0 such that

llexp(tA)--lll~l--8 (ii)

lira

for a l l

t~[O, 6);

Ilexp(tA)--lll=O.

9.1.15. [10] For two given C o -semigroups exp(tA) and exp(tG) the following conditions are equivalent: (i) 11exp(tA) - - exp(tG)II -- o ( t ) as t ---, 0; (ii) D(A) -- D(G) and the operator A is a G - b o u n d e d operator f r o m D(G) into E*. 9.1.16. [369] Let exp(tA) be an OS given on a Grothendieck space E and a) li--m[ICH(t , 1)ll/g
b)

s-lim(tt/t)exp(tA)CH(u, 1)x=o

for all x E E, u > 0.

t~0

Then: (i) the operator

P: =s-(C,

1)limexp

(tA)

is a projector into E, where

~(P)~

t ~

span (Uo~(exp ( t A ) - - I ) )

1108

and

D(P)~{xEE:atn-+ ~o,

n ./P(exp t>O

w-lim n~oo

t21CH(tn,

1 ) x exists};

(tA)--I), W(P)=

(ii) the operator

Qx* = w * - l i m CH(t, 1)* x *

is a projector into E*, where

:~(Q) =

l~o

A~

= s p a n {,U>o~(exp(tA)* - - I*)}

W(A)-~Yl(P),

W(A*)=~(Q),

f] d t ~

_

I),

t>0

and

D ( Q ) = {x*Ee*:at, -+ oo, ~*-limCH(t~, 1 ) * x * exists}. Also,

Yl(A)==,,E'(P), ~ { A * ) = A ~

9.1.17. [369] Let E be a G r o t h e n d i e c k space. An OS exp(tA) is strongly (C, l ) - e r g o d i c if and only if conditions a) and b) of 1.16 are satisfied and w * - - c l ( ~ ( A * ) ) - - - ~ ( A * ) . 9.2. Adjoint Cosine O p e r a t o r Functions. Following 9.1, we let C(t,A) e denote restriction C (t, A)* [E'~, tE R, where E ~ c_ E is a subspace on which the adjoint family C(t,A)* is strongly continuous at zero. 9.2.1. [328] For a C O F C(t,A) defined on E we have (i) if x* E D(A*), then for any t E R

C(t,A)*x*eD(A*) and A*C(t,A)*x*=C(t,A)*A*x* and for any x E E we have t

( x, (C (t, A)* --I*)x* ) = l (t--s) ( x, C (s, A)* A'x* ) ds; 0

(ii) x* ~ D(A*) if and only if the limit

w*- lira (2/s2)(C (s, A ) * - - l * ) x * = 9 * ,

exists, where A ' x * = y*.

s~O+

T H E O R E M 1. [328, 371] In the notation of Prop. 2.1 we have (i) the subspace E | D (A*)~ where closure is taken in the strong topology of the space E*; (ii) the subspace E ~ is invariant under C(t,A)*, and C(t,A) o, t E R, is a C O F on Ee; (iii) the generating operator A | of the C O F C(t,A) ~ is a maximal restriction of the o p e r a t o r A* to E ~ (i.e., A e is the part of the operator A* on Ee); (iv) if E is reflexive, then E e = E * and A ~ = A~; (v) for each t > 0 the operator C(t,A)* is the w ' - c l o s u r e of the operator C(t,A) e. 9.2.2. [328] For any x* E E(A*) we have

II(c (t, A)* - - I * ) x * i,~ < (t2,'2))[I A ' x * il" sup :i C (s, A)ii. 0 ,.s~t

Definition 1. An operator function K(t), t E R, defined on E ~ and satisfying the conditions K(0) = I* and (1.4.1) is said to be a w % c o n t i n u o u s C O F if, for each t > 0, the operator K(t) is continuous in E" in the w * - t o p o l o g y (we write is w" -- w ' - c o n t i n u o u s ) and for any x* E E* the function t --+ K(t)x* is w*-continuous in E* with respect to t E R. 9.2.3. An operator Q E L(E ") is w* -- w*-closed if and only if it is adjoint to a densely d e f i n e d and closed operator ~ 6 L ( E ) . Moreover, D(A) = E ~ if and only if D ( ~ ) = E . , and, in this case, both ~ - and Q are bounded, where IiY'II=I!Qll . T H E O R E b l 2. [37t] In order for an operator f u n c t i o n K(t) on E to be a w ' - c o n t i n u o u s C O F it is necessary and sufficient for K(t) = C(t,A)*, where C(t,A) is some COF. If Q is the generating operator of K(t), t E R (in the sense of the w*-topology) and A is the generating operator of the C O F C(t,A), then Q = A*. We use the notation K(t,Q) for the w*-continuous COF with generating operator Q. 9.2.4. [37] In the notation of T h e o r e m 2, the following conditions are equivalent: (i) x*~D(Q). ( i i ) [ [ K ( t , Q ) x * - - x * [ ] = O ( t 2) as (iii) l i m t - e ! i d f ( t , Q ) x * - - x * ! l <

t-+0;

o~.

t~O+

1109

9.2.5. [371] Let Q ~ L(E*). The operator Q generates a w'-continuous COF if and only if it is w'-densely defined, w * - - W * closed, and there exist constants M > 0 and w >_0 such that for ), > w the point ),2 ~ p(Q) and d rt

9.2.6. [371] If the set _~___D(Q)is w*-dense in D(A) and invariant under the w%continuous COF K(t,Q), then is a w*-core of the operator Q. 9.2.7. [371] For w*-continuous COF's K(t,Q1) and K(t,Q2) the following conditions are equivalent:

(i) D(Q~)~D(Q2)" (ii) II(K (t, Qt)--K ( t, Q2) )x*ll=O ( t 2) for t--~0 for any x*ED(Q1). 9.2.8. [371] The following conditions are equivalent: (i) I[K(t,Q1) -- K(t,Q2)Jl = O(t2) as t - , 0; (ii) D(Q1) -- D(Q2) and Q2 - Q1 is a bounded operator from D(Q1) into E*; (iii) D(A1) C_ D(Q~.) and Q2 - Q1 is a bounded operator from D(Q1) into E'; (iv) D(Q2) _c D(Q1) and Q2 - Q1 is a bounded operator from D(Q2) into E'; Moreover, in these cases, l) Q 2 - Q , il--
Q2) ll--<

(t, Q~)- K (t, Q~) 1i

f>O

where equality is achieved, for example, in the case of contraction w%continuous COF's. 9.2.9. [371] If K(t,Q1) -- K(t,Q 2) = o(t 2) as t ~ 0, then K(t,Q1) = K(t,Q2), t ~ R. 9.2.10. [371] For a w*-continuous COF K(t,Q), t ~ R, we have dl~ --- ,q W ( K ( t , Q ) - I * ) t>0 closure of ~ ( Q ) is the w*-closure U Y~(K(t, Q ) - I * ) . If E is a Grothendieck space, then

, and the

W

-

t>0

(20 (to (t, Q)- z*))= span

Q)- z*):t

where the closure is taken in the strong topology of E*. Notation: t

T(t, A):=I(t--s)C(s,A)ds; 0

Q~:= r

t~co

A)*; t~oo

Q2:=s-limt-~S(t,A)*; ' - - w*=lim t-'S(t, Q~,: t..,-co

(2.1)

A)*.

9.2.11. [371] Assume that for Q = A* the following conditions are satisfied:

a) IIS(t,A)l[

= O(t) as t ~ oo; b) w * A - l i m t - l ( K ( t + s , Q ) - - K ( t - - s , Q ) ) S ( s , A ) * x * = 0 - - : for all x* E E and s > 0. t~co Then (i) Q81 C Qwa - Qw* are projectors, where li Q~.* I[< = liril t-z.It S (t, ,4)[[, while D(Qs 1) C D(Qw 1) and D(Qw .1) t~oo are strongly closed; (ii) ~ ( Q ~ ) = ~ ( Q [ ~ . ) = ~ ( Q ~ * ) =-4~ -/~(Q~)_~A~ ~-~ and IJ/: = s p a n { ~ ( K ( t , Q ) - - l * ) : t > 0 } ~{Q~.)~v* --cI(9~(Q)). If condition b) is replaced by the stronger condition b') s--Iimt-~(I( (t +s, Q ) - t~_x=

1110

K(t--s,Q))S(s,A)*=O

foralls>0,

then L//~=W(Q~).

If we translate Def. 7. 1.4 into the language of COF's and write

P : = s - - lira

t-~S (t,

A ) , we can say that a COF

C(t,A) is strongly (C,1)-ergodic if D(P) = E and uniformly (C,1)-ergodic if IIt - i s ( t , A) - P II ---' 0 as t ---, oo. 9.2.12. [371] Assume that conditions a) and b) of 2.11 are satisfied and E is a Grothendieck space. Then ~(Q~,.)=.ff~(Q), gt~ = ~ ( Q ) and D (Q.~.,) ' =W (Q)~(Q)--E*. If, however, condition b') is satisfied, then K(t,Q) is strongly (C, 1)-ergodic, i.e., D(Qs 1) = E. 9.2.13. [371] A COF C(t,A) defined on a Grothendieck space E is strongly (C,1)-ergodic if and only if the following conditions are satisfied:

(i) [IS(t,A)ll=O(t) for t ~ o o ; (ii)

s-limt-~C(t, A ) S ( s , A ) = O

for a l l

S>0;

t~20

(iii) 7o*-cl ( ~ ( A * ) ) = . ~ (A*). 9.2.14. [371 ] Under the assumptions of 2.13 with a space E that also has the Dunford--Pettis property, the COF C(t,A) is uniformly (C,1)-ergodic if and only if condition (i) of 2.13 is satisfied, I[C(t,A)S(t,A)[] = O(t) for all s > 0 as t --* oo, and, finally, w*--cl(N(A*))=l~A*. 9.2.15. [371] Let II T(t,a)II -- o ( t as t --. oo and s-llrn t - ~ K ( t , Q ) - = O for all x* E D(Q). Then Qs ~ = Qw2 c t --r OO

Q w , 2 a r e b o u n d e d p r o j e c t o r s ( s e e [ 3 7 1 ] f o r t h e d e f i n i t i o n ) suchthat

]iQ~.[i..
_

t --*-Oo

dl~ , "~(Q) =A~176 D (Q2.)=A~

9 :itt. -+ oo : lim

The subspaces O(Qs 2) and n(Qw*) are strongly closed in E" and

2tT2T(tn)x * exists).

n~oo

ooand

9.2.16. [3711 Let E be a Grothendieck space and let K(t,Q) be a w*-continuous COF. If [] T(t,A)[[ = O(t 2) as t ---, "~*-lim t-2K(t, Q)T(s)*x*=O for all x* e E* and s > 0, then ~ ( Q ~ , ) = A ~ A~ D (Q~,) t~oo

~E*.

Moreover, if

s-limt-2K(t, Q)T(s)*=O

for all x > 0, then the COF K(t,Q) is strongly (C,2)-ergodic.

t --r ~

9.2.17. [371] A COF C(t,A) on a Grothendieck space E is strongly (C,2)-ergodic if and only if

(i) liT(t, A)ll-----O(t =) for t ~ o o ; (ii) s-limt-2C(t, A ) r ( s , A)=O for all

$>0;

(iii) w*-cl (N(A*))=N(A*). 9.2.18. [371] Assume that under the assumptions of 2.17 the space E has the Dunford--Pettis property. Then the COF C(t,A) is uniformly (C,2)-ergodic if and only if f[T(t)][ = O(t2), ]1C(t,A)T(s,A)][ = O(t 2) for S > 0 as t -* c~, and

s > 0 , cl (~(A*)) = ~ (A*). 9.2.19. [371] Let

Qx*:=s-limt-lS(t, A)*x*

for those x ~ E E ~ for which the limit exists. The for the w*-

t~O+

continuous COF K(t,Q) we have

D (Q)= U l>0

~(S(t, A)*)={x*EE*

:3tn ~ oo :w*-lim

t21S (tn, A)* x*

exists}. In this

rt--t- o a

case Q = In(G). 9.3. Supplementary Comments. Note that it follows from 1.2 that the adjoint OS exp(tA)*, t E 1/+, is a Co *semigroup, i.e., all of the properties of Co-semigroups hold in the sense of w" properties. However, if we consider arbitrary w*-semigroups in E ' , it turns out that different vC-semigroups may have the same w*-generating operator. The paper [ 171 ] considers the situations in which it is possible to obtain Hille--Phillips type theorems for w*-semigroups, and this paper introduces the notion of integrable w ' - g e n e r a t i n g operators (in this connection, see [ 172, 173, 183,260]). There is an extensive discussion of G-continuous semigroups in the book [10]. We have given only some of the results presented in this book. Concerning adjoint OS's and COF's, see also Props. 4.1.8, 8.t.36-43, and 8.2.17. llll

NOTATION E -- Banach space

.........................................................

L(E) -- the set of linear operators defined on E

1. I

...................................

B(E) -- Banach algebra of continuous linear operators defined on E D(A) -- domain of operator A (A)

-- range of operator A

111.111 g~(A)

-- graphic norm

.................................................

1.1 1.1

................................

1.1

.............................................

1.1

......................................................

-- set of closed operators with

-- the real line (--oo,+oo)

R

1.1

....................................................

-- Banach space of D(A) with graphic norm

or(A) - - s p e c t r u m o f A

D (A)=E

and

p ( A ) 4: r

1.1 .......................

I.I

into E

ck(j;E) -- the set of k-times continuously differentiable

functions mapping J into E

R+ -- the n o n - n e g a t i v e

1.1

..................................................

C ( J ; E ) - - t h e set o f c o n t i n u o u s f u n c t i o n s m a p p i n g j c R

un(t) --n-th

1.i

................................................

p(A) -- resolvent set of operator A ~' (E)

1.1

....................

r e a l s [0,oo)

.......................

1.1 .......

1.1

.............................................

derivative of u(t), dnu(t)/dt n

"

1.2 1.2

9

D - - a s e t d e n s e i n E f o r w h i c h t h e C a u c h y p r o b l e m ( 1 . 2 . 1 ) - ( 1 . 2 . 2 ) is s o l v a b l e

............

1.2

o~ - - a r b i t r a r y c o m p a c t s u b s e t o f R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~"j(t)

6ij

propagators for Cauchy problem (1.2.1)-(1.2.2)

--

the Kronecker

--

............................

delta ....................................................

U(t) -- semigroup of operators

................................................

C o - - t h e class o f s t r o n g l y c o n t i n u o u s s e m i g r o u p s ( O S ' s )

.............................

Banach algebra with identity ~ ...........................................

--

C(t) -- cosine operator function (COF)

..........................................

C(t,A) -- COF with generating operator A S(t) - - s i n e o p e r a t o r f u n c t i o n ( S O F )

.......................................

1.2 '

1.2 1.2 1.3 1.3 1.4 1.4 1.4

............................................

1.4

S(t,A) -- SOF associated with C(t,A) ............................................

1.4

E k - - l i n e a r m a n i f o l d (x ~ E : C ( t , A ) x E C k ( R ; E ) }

1.4

exp(tA) --OS

.................................

with generating operator A ........................................

w(A) - - t y p e o f OS e x p ( t A )

...................................................

R(~,A) = (M ~A) -1 -- resolvent of the operator A (M. a~)

- - set o f g e n e r a t i n g o p e r a t o r s f o r OS's

C -- the complex numbers A '~5

m

....................................

.......................................................

-- core of an operator L?==,~(exp(~A))

...................................

...................................................

- - r e s t r i c t i o n o f o p e r a t o r A to t h e set I)

D(A~176 = f'3kEND(A k )

................................

...................................................

r a n g e o f a n OS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 2.1 2.1 2.1 2. l 2.1

2. t

2.1 2~1

I~, = tJ~>o~ ~ ...............................................................

2.1

N -- the natural numbers

2.1

....................................................

t%(6,F(-)) = sup{ I1f ( t ) - - f ( s ) [I : t - - s < 6, t, s ~ [0,b]} ~o:Z=(exp(~A)--l)/a

...............................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 2.1

r~7

A~-----~, (--1)m-kCmkexp ( k ~ A ) k=0 Cm k -- binomial coefficient

..................................................

2.1 2.1

@(t,x) - - [[ ( e x p ( t A ) - - I)x [[ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

B~=(m!/tm)(exp(tA)--~,

2.1

tkAk/k ) #=0

1112

................................................

............................................

~7=t-m {~o(~l)m-kCmkexp(ktA)x} T m ( t ) = [I m i n (I m i n e x p ( t a ) ) m C 1 -- Cesaro descrete limit ~(M,

...........................................

...............................................

DfA, TI) = {xED(A ~) : ~[ck,,~-----c>O, IlA~x[l
~(0) - - s e c t o r i n t h e c o m p l e x p l a n e kernel of A

0(f~) - - b o u n d a r y o f f~

2.2

>O:]lAhx]J~eaakk,

: Va>0~tc

kEN}

.......

2.2

. . . . . . . .

2.2

.......................................

2.4

............................................

2.4

....................................................

2.4

......................................................

nul ( A ) = d i m d F ( A )

m

dimension of

def(A)

--

codimension

=eodim.~(A)

2.2

: V a > 0 U c > 0 : IlAkx[l~cahk ~, kEN}

{x6D(A ~)

s e t o f g e n e r a t o r s o f a n a l y t i c OS's --the

2.2

...................................

{xED(A |

9 c(A) - - s e t o f i n t e g r a l v e c t o r s o f A , i.e.,

Y(A)

2.1

................................

.....................................................

- - s e t o f a n a l y t i c v e c t o r s o f A , i.e.,

a~(e,o)

2.1

..................................................

co) - - s e t o f g e n e r a t i n g o p e r a t o r s f o r a g r o u p

E X P ( A ) - - U,7>oD(A,rl)

2.1

~(A)

2.4

.......................................

of range

2.5

......................................

ind(A) = (nul(A) -- def(A)) -- index of operator A

2.5

................................

2.5

d i s t ( x , 1 3 E ) = inf{ I] x - - y ]] : y ~ 13} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5

~t(A) - - r e d u c e d m i n i m a l m o d u l u s o f o p e r a t o r A

2.5

SF(A) -- half plane of semi-Fredholmness <x,y')

v a l u e o f f u n c t i o n a l y* a t x

E* - - a d j o i n t o f E

2.5 2.5

.........................................................

Ra(A) -- residual spectrum of A

2.6 2.6

............................................

of A

Ea(A) -- essential spectrum of A

2.5

............................................

Ca(A) -- continuous spectrum of A f~F(A) - - r e g i o n o f F r e d h o l m n e s s

2.5

.........................................

..............................................

Po(A) -- characteristic values of A

2.6

..........................................

2.6

..............................................

Er(B) --essential spectral radius of B IIIBItt~

.............................

............................................

W(A) -- numerical range of operator A

c~(f~) - - K u r a t o w s k i

..................................

of resolvent

2.6

..........................................

2.6

m e a s u r e o f f] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-- measure of noncompactness

2.6

of B E B(E) .................................

2.6

E w ( A ) - - e s s e n t i a l g r o w t h f a c t o r o f OS e x p ( t A ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

Bet(A) - - b o u n d a r y

2.6

spectrum of

AEff (M, co)

......................................

w(x,A) -- bound of exponential growth of exp(tA)x

................................

w l ( A ) - - b o u n d f o r e x p o n e n t i a l g r o w t h o f s o l u t i o n s to C a u c h y p r o b l e m ~ ( x , A ) - - l e f t b o u n d o f w s u c h t h a t t h e f u n c t i o n R ( A , A ) x is h o l o m o r p h i c

2.7

................ when Re A > w

2.7 ....

2.7

"to(x,A) - - l e f t b o u n d o f w s u c h t h a t t h e f u n c t i o n R ( ) , , A ) x is b o u n d e d w h e n R e A > w . . . . . . .

2.7

7 1 ( x , A ) - - l e f t b o u n d o f w s u c h t h a t R ( A , A ) x c o n v e r g e s to z e r o as i m A --* oo, R e A > w . . . . . .

2.7

H ~176 - - set o f f u n c t i o n s f(z) t h a t is b o u n d e d a n d a n a l y t i c w h e n R e z > 0 . . . . . . . . . . . . . . . . . .

2.7

b(x)=lnfIRe~:~e-Ztexp(tA)xdx

2.7

t

exists}

....................................

b

II = 01

.........................................

2.7

Eap = s p a n { x E D ( A ) : A x = Ax, A E iR}

.........................................

2.7

Est = {x ~ E : limt__,oo ]] e x p ( t A ) x E+ - - c o n e

in Banach space E

-- sign of partial ordering sup(S) -- supremum

................................................ ...............................................

o f S in t h e s e n s e o f ~

x+(x - ) - - p o s i t i v e ( n e g a t i v e ) p a r t o f x Ixl-- m o d u l u s o f m e m b e r

.....................................

..........................................

o f a B a n a c h s t r u c t u r e (x + + x - )

E+* - - set o f p o s i t i v e f u n c t i o n a l s

............................

..............................................

2.8 2.8 2.8 2.8 2.8 2.8

1113

D ( A ) + = D ( A ) C~ E + Jx - - p r i n c i p a l

...................................................

ideal corresponding

S ( A ) = s u p { R e X : ,~ E ~ ( A ) } p+(x) -- canonical

2.8 2.8

2.8

.................................................

of Kato inequality

int E+ -- interior of cone E+ Z -- the integers

~. . . . .

..........................................

..................................................

functional

sign x -- linear operator

to x

2.8

.......................................

2.8

..................................................

2.8

..........................................................

2.8

s i g n ( c 0 - - t h e s i g n o f c~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.8

Att -- generator

2.8

tr

~)

of the modulus

-- set of generating

9 piA) - - s e m i a n a l y t i c

9dpp(A)=

CE:

........................................

operators of a COF

we(A) -- type of COF C(t,A) St(A) -- Stieltjes vectors

of an OS

..................................

3.1

.................................................

3.1

.....................................................

vectors

3.1

.................................................

!~Akx'ite/(2k)!<~

f o r all t > 0

.

3.1

. ..............................

3.1

0

N B I o ( A ) = (t > 0 : t h e o p e r a t o r

C(t,A) has no bounded

inverse}

.......................

3.1

NBII(A)

S(t,A) has no bounded

inverse}

.......................

3.1

= {t > 0 : t h e o p e r a t o r

J ( f , e ) - - s e t o f all e - p e r i o d s a(1,f) -- Bohr transform ~t(,~ - - c o m p l e t e

Com(A)

~o(x) - - d u a l m a p p i n g

A-lim 6(t)

9 !(|

--

- - I is a c o m p a c t

.....

...............................

-- the class of Miyadera

7.0 7.0 7.0 7.1

of operator

A

mapping

7.1

................................................ perturbations

8.1

.....................................

operators mapping

E into F

8.1

........................

8.1

.................................................. of E ~

E*

9.1

........................................

9.1

......................................

E ~ - - s u b s p a c e i n E* o n w h i c h t h e a d j o i n t s e m i g r o u p

exp(tA)"

6.0 6.1

....................................................

exp(tA) ~ -- Phillips adjoint of a semigroup

1114

X E ~(A)

........................................................

B(E,F) -- set of linear continuous A* -- adjoint

operator)

to p o i n t o f s p e c t r u m

5.0

...................................................

either an OS or a COF

x** - - c h a r a c t e r i s t i c

5.0

i.e., ( x ~ E :x = f ( t ) , t E R} . . . . . . . . . . . . . . . . . . . . . .

.......................................................

-- Cesaro limit

-- Abel limit

5.0

.........................................................

[x,y] -- semiscalar product (C,~)-lim

...................................

onto root subspace corresponding

= {t > 0 : e x p ( t A ) set

f(t)

of f(t) ................................................

range of a function,

P(,~) - - R i e s z p r o j e c t o r

0(x) - - a d j o i n t

of the function

is s t r o n g l y c o n t i n u o u s

9.1 ......

9.1

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