Two problems on spectral synthesis

7.5.

TWO PROBLEMS ON SPECTRAL SYNTHESIS~

I.

Synthesis Is Not Possible.

We consider the synthesis of the invariant subspaces

for the operator Z* , conjugate to the multiplication ~ :f § zf by the independent variable z in some space of analytic functions. More precisely, let X be a Banach space of functions, defined and holomorphic in the circle

D

these functions, and such that z X c X

, embedded topologically in the space if ~ e X

zero of the function f at the point ~ , ~ ; integral function in the circle

~

, then ~p(~)de~

X~[~:I~X~

~>I ~]

H0~C~)

of all

the multiplicity of the , where k is a nonnegative,

A closed subspace E, invariant relative to Z , is said

to be divisorial in E = Xk for some k [necessarily, ~(g)=~E(~) &el ~ u ~ ~(g),

g6~

].

Conjecture I. In any space X of the described- type, there exist nondivisorial, invar iant Subspaces.

Z-

The statement which is dual to the property of being divisorial consists in the possibility of spectral synthesis for the operator Z ~ : If the space Y, conjugate (or preconju-

gate) with X, is Cauchy realizable [ i . e . , <~,~>= implies

(~)~(nO on polynomials], then Z ' F c F "

E=VCC~-Xz)

z

:~-~-r

(I)

where k = kEi. In other words, the possibility of synthesis for the subspace E means that it is reestablished (in the topology defined by the duality <X, Y>) from the root vectors of the operator Z ~ which are contained in it. All the known results on F -invariance (see [I]) confirm Conjecture I. The principle requirement for the space X is that it be a Banach space and the problem becomes nontrivial if , for example, the polynomials

~A

are contained densely in X and

[~ " ~

sP~~

I pO;)l

•

Jlpn'~
~'_~

) ~I;~ [5]).

This point of view can be given a metric character, can be

connected with the multiplicative structure of analytic functions, with the problem of weighted polynomial approximation, with the generalizations of the corona theorem, etc.; regarding this, see [I, 3, 6]. 2.

Approximate Synthesis Is Possible.

We Shall read (I) in the following manner: There

exists an increasing sequence of Z" -invariant subspaces Era, ~

E~
, approximating

~N. K. NIKOL'SKII. V. A. Steklov Mathematical Institute, Leningrad Branch, Academy of Sciences of the USSR, Fontanka 27, Leningrad, 191011, USSR.

2185

E :E=~E~-~ { 7:~eX, ~ ( ~ , E ~ ) = ~

.

Deleting one word from this proposition, we ob-

tain, apparently, a universal description of Z Conjecture 2. tive to Z* 9

-invariant subspaces.

Let Y be the space from part I; let E be a subspace of Y, invariant rela-

Then, there exist subspaees I n , Z * ~ C ~ #

~g~Em~

~ (~e~)

, for which E =

This conjecture admits a further extension which still seems to be likely. Namely, let T be a linear continuous operator in the linear space Y and assume that the collection of root vectors of the operator T is complete in Y.

Is it true that T I c

~-~[:

"~-E~ , where

~C~,~E~(~)?, It is not true; without additional restrictions on the operator T, it is easy to discover a counterexample: If T is the left shift (i.e., as before Z*!), (a0, az .... ) § (al, ae .... ) in the space IP(wn) with an appropriate (fast and irregularly decreasing) weight {Wn}n~0, then there exist T-invariant subspaces which are not approximable by root subspaces [3]. In these examples it is essential that the spectrum of the operator T should consist only of zero. Many classical theorems on the ~

-invariant subspaces (such as Beurling's theorem) not

only confirm Conjecture 2 but allow us to describe the property

V~Z~: ~0)

X-capacity. ~

[~X

=~

Z

-cyclic vectors f [defined by the

] in terms of an approximation by rational functions with bounded

If r is a rational function with poles in ~\c~o~ ~ , ~(~)=0 , then ~0XZ--~J

: ~Z~, ~> =0,

invariant subspace.

~

; ~(0) = ~I, Similarly one defines also the capacity of any Z ~

If ~ = ~ y > ~

operator ~* ; similarly, ~

r

and ~ X [m

~

~

, then f is not a cyclic element of the

~oo~E~y

. The last statement can be transformed

by a small modification of the concept of capacity [7, 8]. It is possible that the technique of rational approximations allows us to prove Conjecture 2 if we omit (probably, the more complicated) question of the estimation of X-capacities of rational functions (this question is the quantitative form of the uniqueness theorem for the class X). On the other hand, the results regarding this problem, known today, make use not only of the classical uniqueness theorems but also of the explicit description of the Z -invariant subspaces in terms of outer--inner factorizations. LITERATURE CITED I.

2. 3. 4 5.

6. 7. 8. 9.

2186

N. K. Nikol'skii, "Invariant subspaces in the theory of operators and in the theory of functions," in: Itogi Nauki i Tekhniki, Matematicheski Analiz, Vol. 12, VINITI, Moscow (1974), pp. 199-412. I. F. Krasichkov-Ternovskii, "Invariant subspaces of analytic functions. II. Spectral synthesis on convex domains," Mat. Sb., 88, No. 1, 3-30 (1972). N. K. Nikol'skii, "Selected problems of weighted approximation and spectral analysis," Tr. Mosk. Inst. Akad. Nauk, Vol. 120, Nauka, Moscow--Leningrad (1974). B. Korenbljum, "A Beurling-type theorem," Acta Math., 135, 187-219 (1975). S. A. Apresyan, "A description of the algebras of analytic functions admitting localization of ideals," Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, 70, 267269 (1977). N. K. Nikol'skii, "The technique of using a quotient operator for the localization of z-invariant subspaces," Dokl. Akad. Nauk SSSR, 240, No. I, 24-27 (1978). M. B. Gribov and N. K. Nikol'skii, Invariant Subspaces and Rational Approximation (in press). N. K. Nikol'skii, "Lectures on the shift operator. I," Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, 39, 59-93 (1974). H. M. Hilden and L. J. Wallet, "Some cyclic and non-cyclic vectors of certain operators," Indiana Univ. Math. J., 23, No. 7, 557-565 (1974).