Function Theory: Interpolation and Corona Problems (Fields Institute Monographs 25)

N

PJ(z) = I:anp(znI)

=

n=O

I:anznPI(z)

=

J(z)r..p(z) = M
n=O

which by the density of polynomials in H2(l!)), extends to all J Now the operator norm squared of M

E

H2(l!)).

and with J = 1 and M

t~~~l 2~ 127r 1r..p(rei6 )1 2n dO}

1

2n

=

{IIM~III~2(D)} 2~ ~ {II M
I!M",lIop'

which yields !1r..pIiHoo(D) ~ IIM
y = y =

x+We=PC(u-y)+We, or solving for y, (1 + PC)-lpCU+ (1 + PC)-lWe.

The requirements that the internal signals x, v and w be stable (the plant C need not be assumed stable) leads, after a small calculation, to the requirement that F = C(l + PC)-l satisfy the Stability property. i.e.

C(l + PC)-l

E

HOO(l!)).

The compensator C that minimizes the effects of noise is that which minimizes the operator norm

11(1 + PC)-lWll

op

= 11(1 - PF)Wllop ,

since (1 + PC) (1 - PF) = 1 + PC{(l + PC) - 1 - PC}(l + PC)-l = 1. However, factoring P and W into their inner and outer factors P,Po and WiWo (Theorem C.14 (p. 190) in Appendix C), we have from Corollary C.19 (p. 194) (the

12

2. The Interpolation ProbleIIl

following can be made rigorous by assuming P and W are rational) inf

FEH=(D»

11(1 - PF)Wllop

inf

II(Wo - PiPoFWo)Wiliop

inf

IIWo - PiPuFWoliop

inf

IIW" - Pi Gllol'

inf

{IIHII op H(zj) = Wo(zJ)' 1 :::; J < oo},

FEH=(D» FEH=(D» GEH""(D» HEH=(D»

where Z = {ZJ}~l is the zero set of P = <po Thus the noise reduction problem is equivalent to the interpolation problem of finding the bounded analytic function of least norm that takes the values of Wo on the sequence Z of zeroes of P. For a given plant P, solving this problem for all stable plants W can be achieved by solving the interpolation problem (2.4) of Carleson.

2.3 Carleson's duality proof of interpolation We follow Garnett [20] and Seip [43] among otherh. The closed graph theorem (Theorem A 32 (p. 160) in Appendix A) finds many applications within the theory of interpolating sequences, one of which we now describe Given a finite subset Z = {Zj of the open unit disk D in the complex plane C, and a sequence of data ~ = {(J}f=l in C, it is easy to see that there is a bounded holomorphk function J in the disk that interpolates the data, i.e J E HOO(D) and J(zJ) = ~j for

};=1

1 :::; j :::; J. Indeed, the polynomial J(z)

= ~;=1 ~j IT :~"Z is a solution (although i#J

J

,

typically with a much larger supremum norm than necessary). In the 1950's Buck raised the question of whether or not there exists an infinite subhet Z = {Z1}~1 of 11) that is interpolating for Hoo (11), i.e. for every bound~>d sequence ~ = {(j } ~ 1 of complex numbers there is J E HOO(D) such that J(Zj) = (J for 1 :::; J < 00 In 1958 Carleson gave an affirmative answer and moreover characterized all sueh interpolating sequences in the disk. Implicit in Carleson's solution, and explicitly realized by Shapiro and Shieldh in 1961, is the equivalence of this problem with eertain Hilbert hpace a.nalogues which we now describe Let H2(D) be the Hardy space consisting of all holomorphie functions J(z) = ~:'=O anzn in the unit disk with IIJIII-[2(I» = V~:=o la n l2 < 00. The Hardy space H2(D) can be identified with the closed subspaee 1-£2('11') of £2('11') given by

Indeed, simply associate each J(z) integrable fUIlction

=

~:'=O a"z" in the unit disk with the square

J* on the circle satisfying

Pen) =

{ao'

~~~ ~ ~ ~

. The

inner product on H2(D) is then defined by that on 1-[2('11') inherited from £2('11')

2.3. Carleson's duality proof of Interpolation

13

if I(z) = 2:~=oanzn and g(z) = 2:~=obnzn. Cauchy's Theorem applied to polynomials, and then followed by a limiting argument, f:ohows that

~

I(z) =

2m

(

f*(e i9 ) d(e i9 ) =

Ie e,9 -

Z

~ 271"

f*(e~9)

(

Ie 1 -

e .9 Z

dO = (f, kz)

where the "reproducing kernel" k z E H2(lIJl) satisfies kz«() = l!z( and k;(ei9 ) = ,g. We note in passing that if I(rkz) is a rapidly converging sequence in H2(D) (where the radii r k tend to 1 from below), then f*(e i9 ) = limk-+oo l(rkei9) for a e. O. Moreover, Fatou's Theorem shows that we actually have f*(e i9 ) = linlr -+ 1- l(re i9 ) for a.e 0, but we will not need either of these facts for the moment. The above computations show that for fixed ZED, the linear functional I -> I(z) is continuous on H2(lIJl) with norm Ilkzll = v'(kz , kz) = v'kz(z) = (1-lzI 2)-L

l-i..

Thus the map I -> {/(z)V1 -Iz) 12}~1 is bounded with norm 1 from H2(lIJl) to eOO(Z). The relevant quef:otions that Shapiro and Shields then asked were these. When is this map onto (respectively into) the smaller Hilbert space £2(Z)? In other words, when is the ref:otiction map RI = {/(Zj)}~l onto (respectively into) the weighted space

£'(p)

~ {~~ {~;}~, ' 11,11,,(,) ~ ~ 1,;1' "; <

00 }

with weight J.Lj = 1-lz)1 2 = k,)~zj)? We will see in a moment that these questions are equivalent to Buck's question above. But first we show how the closed graph theorem yields an unexpected control in the way we may interpolate the data ~ E e2(J-t) when R maps H2(lIJl) onto (but not assumed into) £2(J-t). Lemma 2.1 Suppo.~e that £2(J-t) C R(H2(lIJl» Then there is a constant C such

that for every ~

E

£2(p). there is f

E

H2(lIJl) satisfying

1 Rf =~.

2.

IIfIIH2(1I)

~ C 1I~IIR2(~)'

Proof. Let M = {J E H2(lIJl) fez)~ = 0, 1 ~ J < oo} Then M is closed since point evaluations are continuous, and we have H2 (lIJl) = M EEl M 1. by Theorem A.10 (p. 150). Let p1. denote projection of H2(lIJl) onto M 1.. Then if ~ E £2(J.L) and fi; E H2(lIJl) satisfies Rli; =~, we have that Rp1. fi; = ~ with p1. Ii; the unique such element in M 1. Thus A~ = p1. It. defines a linear map from £2 (J.L) to M 1. with RA~ =~. Clearly A hm. closed graph since if (e',A~k) -> (~,h) in £2(J.L) x M1., Le. ~k -> ~ in £2 (J.L) and A~k -> h in M 1. , then both

~; ->~) and A~k(Zj)

->

h(z)

-> 00 for each fixed J. However A~k(Zj) = (Rp1. ft."'») = ~j then shows that h(z) = ~) = A~(zj) for all J and hence by uniqueness in M1. that A~ = h, Le. (~, h) is in the graph of A. The closed graph theorem (Theorem A.32 (p. 160) in Appendix A) now implies that A is continuous and we may take I = Ae in the

as k

conclusion of the lemma. An even easier argument using the closed graph theorem shows that if R maps H2(D) into £2 (J.L), then it does so boundedly

2. The Interpolation Problem

14

Before stating Carleson's Interpolation Theorem, we derive an interesting consequence of the interpolation control given by (2) in the lemma above. Corollary 2.2 If £2(1-') C R(H2(lJ))) then Z = {ZJ}.i=l is separated in the Poincare metric: there is a positive constant c such that

IZi - zjl ~ cmin{I-l zil, 1 -Izjl},

i

#- J.

Proof: Fix j and let ~ satisfy ~i = o~. By the lemma there is

f

E H2(lJ)) such

that f(Zi) = o~ and IIfIlH2(J[)) ~ C 1I~lIl2(1') = CJl - IZjl2 with C ~ 1. Thus for i #- j and any complex scalar >.., we have using the reproducing kernel kw(z) = l-~Z'

(f(zJ) - >..f(Zi))2 = (I, kZj - >"kZi /

1

< IIfl\~2(][})

(1 - IZj 12) {llkzj 1I~2(J[)) + 1>..1211kzi \1~2(J[) - 2Re (kzj , >"kz.) }

< C

C (kz j C

if we choose

,

A

1

Ilkz] - >"kz.II~2(][})

~Zj)) { kZj (Zj) + 1>"1 2 kZi (Zi) - 2 Re >"kZi (zJ) }

{2 _2Jk

=

Ik zi (zj)1 } zj (Zj) Zi (Zi)

Jk

Ik •. (Zj)1 k. Zj U· h'd . ') ( ) . smg tel entIty k •• (Zj k •• Zi \ Zi - ZJ

[2

l-zjzi we have 1

and hence

-

(1-lziI2)(I-\ZjI2) Ikzi (zJ)\2 11-zjzil2 = kZi(Zi)kzj(zJ)'

-I

Zi - ZJ 12 1- ZjZi

(2.6)

~ (1 _ ~ )2, 2C

I:~:;;;. I~ c > 0 for all i #- j, which easily yields the corollary.

ReIIlRrk 2.3 The useful identity (2.6) arises very often in function theory on the disk, and generalizes to other situations, e.g. (6.19) on the unit ball. We note that if R maps HOO(l!)) onto lOO(Z), then the open mapping theorem easily shows that Z is separated. Indeed, there is c > 0 such that R (unit ball in HOO(l!))) :J cxunit ball in £OO(Z). Thus for fixed i #- j there is f E HOO(l!)) of norm at most one with f(Zi) = c and f(zJ) = O. If CPw(z) = l~~z is the involutive automorphism of the disk l!) that interchanges 0 and w, then 9 = ! 0 CPz. maps l!) to l!) and 9(0) = !(Zj) = O. Thus the Schwarz lemma implies that (recall CPZj 0 CPZj is the identity) ]

which shows as above that Z is separated. Thus separation is a necessary condition for either £2(1-') C R(H2(l!))) or £OO(Z) C R(HOO(lJ))), and we may as well assume it from the outset. Now we can state Carleson's Interpolation Theorem with contributions from Shapiro and Shields.

2.3. Carleson's duality proof of interpolation

Theorem 2.4 Suppose that Z

15

= {Zj}~l

C

JDl is separated, i.e.

I ?:. c > 0 for all i i= j, I1Zi- ~j ZjZi let R be the restriction map Rf = {f(zj)}~l and f.1z the following conditions are equivalent: 1. R maps HOO(JDl) onto £OO(Z), 2. R maps H2 (JDl) onto £2 (f.1 z), 3. R maps H2(JDl) into £2 (f.1z) , 4. f.1z(T(I) :'S C III for all arcs leT. 5. infj

IT iofj

(2.7)

= 2:}:1(1-lzjI2)8zr Then

I t~;:~ I == 6z > O. J

,

Here the tent T(I) over the arc I is the convex hull of I and ZI = re ifJ where eifJ is the midpoint of I and r = 1 - ~~. Marshall and Sundberg identified the crucial interplay between the spaces HOO(JDl) and H2(lIJ) here, namely that HOO(lIJ) is the multiplier algebra MH2(JfJJ) of the Hilbert space H2(lIJ). This will be exploited in the proof below Definition 2.5 If R maps HOO(lDl) onto £OO(Z) with kernel N, then Ii : HOO(lDl)/N ---4 £OO(Z) is invertible by the open mapping theorem (Theorem A.29 (p. 158) in Appendix A). Denote by Mz the norm of the inverse map.

The proof below will show that

~ 6z

< Mz < C~ 8z

(1 + ln~) 6z

.

There are examples on page 284 of Garnett's book [20] of finite sequences Z with M z ~ (j~ In (j~ for arbitrarily small 6z > O. Remark 2.6 We have stated the theorem for the unit disk lDl, but in the course of the proof we will switch back and forth between the disk and the upper half plane 1U, using the setting most convenient for a given argument. In fact, arguments involving 5 are easiest in (a bounded region of) the upper half plane, while arguments not involving 5 are better handled in the disk. Here is the reason. Using the Cayley transform Z = i that takes lDl one-to-one and onto 1U, we see that the Blaschke product in l!J with zeroes Z = {i, ... , i} U {Zj }~1 C 1U\ {i}, where i has multiplicity k ?:. 0, is given by

i+:

B(Z)=(Z-~)krrOO Z + 2 J=l .

!l+z;!z- z

J,

1 + zJ~ Z - Zj

and that the corresponding Blaschke condition is 00

'~ " j=l

where

Zj

=

Xj

+ Yj·

YJ 2 1 + IZjl

< 00,

In the special case that Z is bounded, then the Blaschke 00

condition becomes simply 2:}:1 Yj

< 00, and the infinite product IT j=l

:=:' converges

uniformly on compact subsets of 1U without the compensating factors

J

I~::~

I. Thus

16

2. The Interpolation Problelll

we can redefine the Blaschke product in this case to be simply B(z) =

00 n :=~, 3=1

3

and with little loss of generality we will do this in the proof that follows. Proof. Conditions 1, 3, 4 and 5 are equivalent in more general situation~ where 2 is no longer sufficient Thus we begin by showing the implications 1 =?5 -{=> 4 -{=> 3 =?- 1 00 (1 implies 5) Let B(z) = ~=~ be the Blaschke product in the upper half

n

j=l

J

space llJ associated with Z and set

B;(z)

Z-z

= __ J B(z) z - Zj

By the open mapping theorem, the one-to-one map f --+ {f(Z)}ZEZ from HOC IN onto £00, where N is the kernel of the restriction map, has a bounded inverse If Afz is the norm of this inverse, and fJ E HOC solves fJ (Zi) = 8j with IIf} \\00 :=::; Mz + c, c > 0, then by Corollary C 7 (p 187), 9} (z) = ~~~~) satisfies jig} 1100 we obtain

\Bj~Zj)\ = I~;~;/)I = \9;(Zj)\ ~ \\9jIl00 = > ° Thus {}z = inf; \Bj(zj)\ ~ Jz > °

IIfJlloc

= II Ii \\00

and

~ M z +€,

for every € (5 implies 4) We continue to work in the upper half plane First we ~how that 5 implies the following restricted Carleson condition for the associated measure p, = p,z = L~l YjOz j where zJ = Xj + iYj·

p,(T(Zk» =

L

Yj:=::; CYk = C x height of T(Zk),

(2.8)

zjET(z.)

where the tent T(Zk) is the equilateral triangle with vertex Zk and opposite side on the x-axis Fix k. If Bk(Z) = TIj #k ;=~ denotes the Bla..o;chke product with zeroes Z \ {Zk}, t.hen from - In t ~ 1 - t for t > (which follows since In is concave and t - 1 is t.he t.angent line to the graph of In at (1,0», we have that

°

2ln

~ Oz

> -In\Bk(zk)\2 = - LIn 1Zk - ZJ 12 Zk -

}#k

>

L

)#

(1 -1

Zk Zk

(2.9)

z)

=ZJZJ 12) = LNk \Zk4Yk~ - z,\

2

For future reference we note the converse inequality -In \Bk (Zk)\2 :=::; (1

+ 21n ~) L

4Yk~ 2' z)\

c 3'# \Zk -

where c is the separat.ion constant in t.he upper half plane analogue of (2 7)

I

z1 -

Zi

Zj -

Zi

I~

c

> 0,

i

i- J.

This follows from the following reverbe tangent line inequality -In t

~

(1 +

2ln ~) (1 - t)

for c2 < t <

1

(2.10)

2.3. Carleson's duality proof of interpolation

Now if ZJ E

T(Zk),

IZk - Zjl2

~ 5y~ and so,

~ Yk I:

5YkY~2 ~

then

L

YJ

17

ZJET(Zk)

(3 In 81) Yk

zJET(Zk) IZk - zjl

z

To finish, a covering lemma shows that we have the extended inequality, p,(T(z»

L

= ZJ

Yj

(4

~

8~) y,

In

for all z

= x + iy.

(2.11)

ET(z)

To see this, fix Z = x + iy, let I Zj be the ba.~e of the tent T(zj), and let {IzkhEE be a subcollection of the intervals {IzJ . zJ E T(z)} having union UZjET(z)Iz" and finite overlap 2 (we may assume the sequence finite for this). Then

2:

2: 2:

<

YJ

zjET(z)

~ 2: (3 In 8~) Yk

Yj

kEE zJET(zd

kEE

31n ~ 8 '"' ~ v'3 2

=

Z kEE

(31n

11 I <31n ~ 8 v'3 2 11z I Zk

-

z

o~) Y

l'z~T(I» we have proved CP.z < 1 Thus' WI·th C1'£ -= sUPICTII - CIn oz' (4 implies 3) Now we work on the disk Given f E H2, which we view as the Poisson integral of its boundary values f* = f, let N f(e iO ) = sUPzEro If(z)1 be the nontangential maximal function where ro is the convex hull of {e iO } U !lD>. Then

from Theorem C.lO (p 187) we have 0),

IT INf(e i9 )12 dO ~ C 1/f11~2'

= {O' Nf(e i9 ) >

A}

Moreover,

= U~II(OJ,rj)

where the I(Oj, rJ ) are the maximal open intervals in the open set 0), and ICO, r) = {ei 10 -
{z. /f(z)1 > A} C U~ITz3 together with the

C(trle~on

('ondition 4 yield the DC

p,z( {z If(z)1 > A}) ~

2: P,z(T

z ,)

~

inequality,

2: Cr

J

=

C I{O : N f(e i9 ) > A}I·

j=1

J=1

Combining inequalities yields

di~tribution

00

1 1I 00

l/f(z)1 2 dp,z

< C

P,z( {z . If(z)1 > A} )2AdA

00

(O : N f(e iO ) > A} 12AdA

C LIN f(e i9 ) 12 dO

< Thus with

C.JCp.z

IIlLzIIHLCar ==

sUPNO

Cllfll~2

I1fl:lli~:'Z),

we have proved that

IllLzllHLCar

~

18

2. The Interpolation ProblelD

(3 implies 4 implies 5) Clearly 3 implies 4 in the disk upon testing over the reproducing kernels kw(z) = 1-;"20 of H2 and noting that

Clkw(z)1 ~

1

1-lwl

2

= kw(w) =

2

II k wllH2

for Z E Tw Indeed, the closed graph theorem shows that R maps H2 boundedly into t2(p.), and we then conclude that

,.(Tw )

C'~~'

)'

<;

!.1k",(z)I' d,.(z)

<;

CUk", II~, ,

or p.(Tw) :-::; C II k w Ii H; = C(1-lwI 2 ) for all wED, which is 4. We have shown that C IIp.z lI e ar·

...;c;:; :-: ;

Remark 2.7 The arguments used here to prove the equivalence of 3 and 4 extend immediately to more general positive measures p. to prove a theorem of Carleson: the embedding H2 C L2(p.), i.e. for all h

E

H2,

holds if and only if for all arcs leT.

p.(S(I)) :-::; C III,

Sn

Now we show that 4 implies 5 in the upper half plane. Fix E 1U : Iz - xkl :-::; 2n Yk}. Then from 4 we get

Zk = Xk

+ iYk.

Let

= {z

00

< 8C+32CI:T n

:-::;

C.

n=l

By (2.10) we then have

which yields 5. (5 implies 1) We are in the upper half plane. It suffices to assume that Z = {Zl' ... , zn} is finite and obtain estimates independent of n, i.e. depending only on n

5z . Recall that B(z) =

II

:=:~ is the Blaschke product associated with Z and

j=1

Bj(z)

=

:=:3 B(z). Note that 3

B'(Zj)

=

Bj(Zj) . ZJ - Zj

Then a (usually non-optimal) solution to the interpolation problem

J(Zj) =

aj,

1 :-::; j :-::; n,

J E H oo ,

(2.12)

19

2.3. Carleson's duality proof of Interpolation

is given by

~ Bj(z) n ~:::~j B(z) ~ B(z) z - Zj L-aj Bo(Zo) = L>j B'(Zo)(Zo - ZJ) = ~aj B'(Zj)(Z - Zo) Zo - Zo'

J=1

J

J

j=1

J

J

J=1

'J

,

and so also by

B(z)

n

~aj B'(Zo)(Z _

g(Z) =

J=1

Zo)"

J

J

If I E Hoc is another solution, then 1-9 vanishes on Z, so equals Bh for some h E HOC by Corollary C.7 (p 187) Thus the minimal HOC norm of a solution I to the problem (2.12) is given by inf

hEH=

119 + Bhli Hoo

(t,

B'(Zj)~Z _ z;) + h(Z))

.!ri1-

B(z)

h~r;foo

La, B'(zo)(z _ Z ) + h(z)

a;

1

n

J=1

)

J

L=(R)

Now for any Banach space X with closed subspace Y, Proposition AAO (p. 165) in Appendix A (which uses the Hahn-Banach Theorem) gives the isometric isomorphism Y* ~ X* /y.i, i.e. for x* E X*,

+ kilo k E y.i} = £1 and Y = HI, then

inf{llx*

sup{l(x*,y)lo y E Y, Ilyll:::; I}

If we take X = y.i = HOC and we have (without using the axiom of choice here since Ll is separable),

.

mf

hEH=

119 + Bhll Hoc

=

~1

sup

L-

fEHl,lIfll9 )=1

R

aJI(x) B'()( Zj X -

) dx . z)

Now the definition of the interpolating constant Mz (Definition 25) and an application of Cauchy'S Theorem give

Mz

=

sup

sup

lI{aj}lloo~1 fEHl,lIfll9 2rr

sup

t

fEHl,lIfll9 )=1

4rr

sup

t

fEHl,IIfll9 j=1

~ a I(Zo) 2rr 6 ), ) j=1 B (Zj)

II(z))1 IB'(Zj)1 Yj

II(Zj)1

IB)(Zj)1 '

since IB'(zj)1 = IB~~:j)1 If we let Vz = Ej=1 the norm of the embedding HI C Ll(vZ), i.e

IBj(~I)18zj' then we see that Mz is

20

2. The Interpolation ProbleIIl

Now by 3 (already proved) the norm of the embedding H2 C L2 (IL z) is bounded by C Jl + In li~. Now we use the factorization of I E HI as I = gh with g, h E HZ and II/l1Hl = IIgllH211hllH2 (Corollary C 8 (p. 187) in Appendix C) to obtain

fu I/(z)1 dlLz

1

<

(fu Ig(z)\2 dIL (Iv Z) 2

< CJl + In C ( 1 + In

1

Ih(z)1 2 dlLz )

:2

<5~ 119!!H211J1 + In <5~ hl!H2 <5~ )

IIIII HI

This yields IIlLzlIHl-car :=:; C(1 + In li~) and so

Mz

= II v zllHI_car

1 1 :=:; C <5 z (1 + In <5z)·

Remark 2.8 Instead we could have appealed to Neville·s proof in the disk Note that with 9 = BI we have

g' = B'I+B/', and so

j=l

where U (Zj) denotes a hyperbolic ball centred Zj with small radiUb Using the mean value theorem for holomorphic functions we obtain for a slight enlargement of the ball U(Zj),

[ III dlLz :=:; J[)

~

z

t[

j=l JU(z/)

I(Bf)'(z)1 dz:=:; [\(Bf)'(z)1 dz

J'J}

Now using (1.1) and the embedding H1(1I.') C B{(Jl))) we have

L

I(Bf)'(z)\ dz + IB 1(0)\ = II B III B i(ID) :=:; II B III 11 1(1I') :=:;

c II/I1 H I(lI') ,

which shows that IIlLzllHLcarleson :=:; <'i~, an adequate but crude estimatE' It remains to show that condition 2 is equiv-dlent to the others \Ve work entirely on the disk for the remainder of the proof. To see that 2 is &ufficient, we prove the strong separation condition 5. Fix J and by Lemma 2 1 <:-hoose I to be a solution to RI = 8J(i) with !lf1lH2 :=:; C 118JlIl2{l'z) = C(1 - IZJ\2)!, where

ILz =

Z::;1 (1-lzj!2)8

zj

With Bj as above, Corollary C.7 shows that g(z) =

£f(2)

21

2.3. Carleson's duality proof of interpolation

satisfies l!g!lH2 = II/IIH2 and 1

I

IB)(Zj)!

SO

BI)«zzj3~) 1 = Ig(Zj) I = !(g, kZj ) I

< IIgIIH2 !!kzJ IIH2 = 1I/IIH2 (1 -lz)1 2)-! :.:; c, which is 5. For the nece:::.sity, it will be convE'nient for later purposes to introduce two further function theoretic propertie:::. of a sequenC'e Z We say that Z = {Zj }~1 batisfies UBS if the reproducing kernels kZ3 form an unconditional basic sequence in H2 oc

00

whenever Ibjl :.:; lajl

:.:; C Lajkz ) )=1

We &ay that Z ba,::,i& in H 2 .

Hl

(2.13)

H2

3=1

= {Zj }i=1 satisfies RB if the IeproduC'ing kernels kZj form a Riesz

H2

;: : (f

la jl2 )

~

(2.14)

)=1

We now show the ncces&ity of 2 by an abstract approach. We pause momentarily to identify the multiplier algebra MH2 of H2. By definition MH2 = {'P . IIJ) -+ C .
U,M~kll') = (
1
11,p(W)k1111i = I!M~kwll :.:; I!M~lllIkwll,

i.e 1'P(w) I :.:; !!M~I! = l!Mcpll, and..p E Hoc. Conversely, the definition of H2 shows that lI
= Hoo with equality of norms.

First, we have that UBS follows from 1 using M~(kzj) we choose

00

:.:; IIMcpll Lajkzj j=l

H2

=
22

2. The Interpolation problem.

Next we show that UBS implies RB. For this we use the fact that for any finite collection of vectors {Vn}~=l in a Hilbert space H there is () E [0,211") such that

lite'' v·II' ~ t IIv.II'

(215)

Indeed, we simply compute the expectation,

2~

tilt e'''''v.1I'

dJJ

~ ~t (v~, v.l 2~ t

e'(--')' dJJ

t

~ IIvnll' ,

and use the intermediate value theorem. From (2.15) we thus obtain

and

[1~(;n8{e''''''') IIk~:"Hl, :; c[[t,e;"""lIk~nH' II:. .

Now let N ~ 00 to obtain RB. Finally we prove that RB holds if and only if both 2 and 3 hold. In fact we prove more: 2 is equivalent to ~ in RB, and 3 is equivalent to ~ in RB. The arguments are short and essentially reversible. First, conditions 2 and 3 are trivially equivalent to the map f - t Tf = {IIL~zlt2 }~l taking H2 onto respectively into (J.. If the map T is into, then we have

sup

f

J(Zj) a-

IIfllHFl 3=1 IlkzJ IIH2 3 <

IIf1lsup H2=1

(f [llt(I') [2)! II{a3}~11If2 j=l

Zj

H2

< CII{aj}~11!£2' which is ~ in RB. If the map T is also onto, then by Corollary A.39 (p 164) in Appendix A, its adjoint T*, given by

T*({

}OO)

aj j=l

satisfies

~-

= ~aj 3=1

Ilk I H2 ' kZj

Zj

2.4. Peter Jones' constructive proof of interpolation

23

which gives the opposite inequality in RB Conversely, if the inequality ;S in RB holds, then =

sup lI{aJ}~,llf2=1

sup

II{aj}f=,lle2 =1

< If the reven,e inequality

~

in RB also holds, then

IIT*( {aj}~1)IIH2 =

f

aJ

Ilk kZ11

J=1

ZJ

H2

H2

~ cII{aJ}~111.~2 ,

which, by Corollary A.39 again, bhows that T is onto. This completes the proof of Theorem 2 4. 2.4 Peter Jones' constructive proof of interpolation We follow the excellent exposition in Seip [43] to construct Jones' linear interpolation opeIator. We wish to construct bounded analytic functions fJ in the upper half plane such that (2.16)

L

< C < 00,

Ifk(Z)1

for all z.

k=1

With this done, the function f = I:~=l ak/k(z) takes the value ak at Zk and has Ilfll H= ::; {aj }~llloo' showing that Z is an interpolating sequence for the upper half plane A natural choice to try first b

ell

Bk(Z) /k(z) = Bk(Zk) ' since fk E Hoo batbfies the first condition in (216), fk(Zj) = 8J ,k' However, the Blaschke pJOducts have modulus one on the boundary JR (again it suffices to consider only finite sequences Z), and so typically I:~1 Ifk(Z)1 ~ 00 as Z ~ JR. To remedy the problem at infinity, we introduce the bounded holomorphic function g(z) = (z+';)2 that vanishes at 00 (note g(i) = 1) and try the modification

fk(Z)

Bk(Z)

= Bk(Zk)9k(Z),

where gk(Z) = g( Z;:5) = (z-=-4:k~'2 is 9 rescaled and translated to take the value 1 at Zk. Of course the modified /k's still fail to satisfy the second condition in (2.16). This could be fixed by further modifying the fk by multiplying by an exponential factor of the form e - Q E Yj :SYk

I:t/::) I, but this has the defect that it

2. The Interpolation problem

24

fails to be holomorphic. Instead, we use a positive harmonic majorant the corresponding exponential factor

U

for

Igl and

where v is the harmonic conjugate of u in the upper half plane. Here are the details Let u(x, y) = x2~(t':J)2 = ~~~~~ be 4 times the Poibson kernelfor the half space R x (-1,00) so that u is positive harmonic and coincides with Ig(x y)1 = x2+(1+y)2 on the boundary y = O. Obviously, u ~ Igl (or more generally by the maximum Principle) Define Uk(Z) = u(Z-XA.) = 4111,('1/"+1) and Iz-zd

Yle

~

Un(z)

Uj(z)

L...J IBj(zj)l·

=

vj'5.vT<

Note that Un(z) is finite, and so a positive harmonic function, by the extended Carleson inequality (2.11) with z(l) = x + i2£Y:

L

c- 1

L

4yj(yj~;)+C-1f Iz -

z J ET(z(2»

Zj

1

c

-1

Iz -

£=2 zjET(z(l»\T(z(l-l»

~S~

< C

4YJ(YJ~;) ZJ

1

~Sh

oc

Y.. + Y

~

. + C -1 ~ YTI + Y YJ C L...J (2£)2

Y

zjET(z(2»

L...J

2

l=2

y

~

L...J

YJ'

zJET(z(£»

Yj"SY"

< C

C

-1

~ Yn + Y L...J (21)2 £=1

(41

Y

< C (ln~) Yn +y < C C Y

n

~) 2£ C Y

00,

but that this bound is not uniform in Z and n With Vn the harmonic conjugate of Un in the upper half plane and G n = Un + i\f,.., we set

In(z) =

:nn(~~)gn(z)e-a{Gn(z)-G,,(zn)},

for a positive constant Q to be chosen momentarily. Clearly, the In are holomorphic in the upper half plane and satisfy the first condition in (2.16). We estimate the second condition in terms of the quantity C 1 6(Z) = supUn(zTI) :S -Inn;?l

C

C

to obtain (!!) oc

Llfn(z)\ n=l

(2.17)

25

2.4. Peter Jon_' conatructive proof of interpolation

with a = "'(Z)' since if the Zj are ordered with decreasing Yj and tn for any fixed z, then

= L: Yj :-:;Yn 1~:tt~1

Thus we have obtained (2 16) with constant C = e /'::,. (Z) (see [43] for a discussion of sharpness). 2.4.1 The Carleson measure d-bar equation. Using the above linear operator of interpolation, Peter Jones [22] constructed a solution u to the d-bar equation

au az(Z) = dJ.L(z),

(2.18)

that is bounded on JR. when J.L is a complex measure such that 1J.L1 is a Carleson meabure 1J.L1 (T(z)) ~ Cy for all Z in the upper half plane. In the next subsubsection a variant of this equation arises in Wolff's proof of the corona theorem. Here we :,ketch the connection between (2.18) and interpolation. Let Z be an interpolating sequence with corresponding Blaschke product B (i e. Z is the set of zeroes of B). With hn(z) = Z - ~gn(z)e-a{G,,(z)-G"(ZTL)}, Zn -Zn we can rewrite the above linear operator of interpolation as

t;

=

t; co

00

fez)

ajfJ(z)

= B(z)

:z z!z _

Green's Theorem shows that

hj(z) aj B'(zj)(z - Zj)·

(2.19)

= m}< in the distribution sense (see (3.22)

J

below and note that d( 1\ d( = 2id{ drJ), and a straightforward calculation now showb that the function u = ib a bounded (on JR.) solution to the equation

Ii

au _ 1 ( a . a _ .~ ajYJ = = -2 -;:;- - t-;:;-)u - 27ft L,; B-( 0)8 z;uZ uX uX j=l J zJ To handle the complex Carleson measure J.L in (2.18) we essentially write J.L as a limit of convex combinations of measures J.Lz = L:j:l y j 8zj associated to interpolating sequences Z = {zJ }f=1. The connection is that a separated sequence Z is interpolating if and only if J.Lz is a Carleson measure Here are the details. First we may assume that J.L in (218) is a positive Carleson measure with

!lJ.Lllcarlf'son ==

~«QQ)) ~ L

sup

Q dyadic square on IR {.

Now approximate J.L by choosing T, c > 0 small and N < 00 large, discarding the mass of J.L outside the compact set Lr = {z . Y 2': T and lzi ~ ~}, dividing the upper half plane into "hyperbolic squares" K J of diameter c, and then replacing the IIlasS of J.L on Kj by the point mass ~YJ8:Z;j where Zj is the center of K j , o ~ k j ~ N 1IJ.L!l carleson = N and

!J.L(Kj ) - ;Yj!

~ :~.

2. The Interpolation probleDl

26

The resulting positive measures

converge to /-t weak star provided r, £ variation of the error measure Wr,E,N

-7

0 and N

-7

00

in such a way that the total

==

satisfies, with dA(Z) invariant measure, 1

A(L.)

::; #{J : K J n L. -=I- 4>} 2Nr ::; C 2Nrc 2

II Wr,f:,NII =

-7

O. (220)

Indeed, the natural approximation /-tr,c,N

L

=

/-t(K)8zJ

j KjnLd'

obviously tends to /-t weak star while the error W.,f:,N tends to 0 strongly MOIeover, the measures /-t.,f:,N have Carleson norms bounded by 2 if IIw. c.NII ::; r. Indeed,

/-tr,f:,N(Q) ::; /-t(Q}

+ W,.f: N(Q)

::; £(Q)

+ W" c,N(Q} ::; 2£(Q),

since /-tr,f:,N(Q) -=I- 0 implies that £(Q) ;?: r. Momentarily fix such a meabure v = /-tr,f:,N with (2.21) (which using (2 20) can be accompli&hed by choosing N laIge enough depending on £}, and note that v has the form

r and

where 0 ::; kj ::; N. Note also that v(Q) ::; 2£(Q) for all dyadic cubes Q on JR.. Consider a new sequence {(j}f=I' such that each point in Z = {Zj}f=l occurs exactly kJ times in {(j}f=1 Then with (j = ~1 + irJj we have 1 00 v = -N~ "rJ·8r )"3 )=1

We now decompose the sequence {(j}f=l into 6N subsequences Zl, . ,Z6N each of which has separation constant at least ~ and Carleson norm at most 4. For this we first note that if T( Q) denotes the top half of a square Q resting on the x-axis, then

#{j : (j

E

T(Q)} ::; 3N.

(2.22)

2.4. Peter Jones' constructive proof of interpolation

Indeed, we may assume that l(Q)

#{J : (j

E

T(Q)}

~

27

r, and we then have

=

where the final inequality is (2.21) Order the points (J = ~j + iTJJ (with the necessary repetitions) in the strip Sn = {( = ~+2TJ' 2- n - 1 < TJ:::; 2- n } accordingtonondecreasingrealcomponent~j' and partition them into new sequences Yt, 1 :::; £ :::; 2N, by putting every 3Nth point in order into a given Yt. From (2.22) we see that the points in YtnSn have separation constant !. Thus if we define Z2t = Un evenYt n Sn and ZU-l = Un oddYt n Sn for 1 :::; £:::; 2N, then each of the sets Zk has separation constant at least ~. It remains to show that the measure f-£YI = L(,EY, TJjb zj corresponding to Yt has Carleson norm at most 4. However, if Q is any dyadic square of side length £(Q) resting on the x-axis, then we have /-Ly,(Q)

L L

=

TJj

<

L

=

(,EYtnQ

L

TJj

2-"9(Q) (jEY,nQnS"

#(Yt n Q n Sn)2- n

2-n~;t(Q)

(1 + #(~~ Sn)) 2- n

L

<

2-"~i(Q)

1

< 2£(Q) + 3N

L

2TJj

(jEQ

2£(Q)

+ ~1I(Q)

:::;

(2 + ~)

£(Q)

Thus each subsequence Zk has separation constant at least! and Carleson norm at most 4, and so is an interpolating sequence for HOO(~) uniformly in k, N, r and IS Now apply Jones' linear operator of interpolation (219) to the sequence Zk = {zj}~l' zj = xJ + iyj, with aj = 2~iBj(zj) to obtain a bounded (on R) solution Wk to the equation ~ = f-£z • = L'EZ y~bzk, with bound uniform in k, J A. J j N, r and

IS

1 ",,4N L.,k=l Wk

Then Ur,e,N = Iii

solves

a

crzUr,e,N(Z) = d/-Lr,e,N(Z) with uniform LOO(R) bounds. Now we wish to make sense of the limit U = limr,e->o,N->oo Ur,e,N as a solution to (2.18) in some suitably useful way. Of course we will let r, IS -+ 0 and N -+ 00

28

2. The Interpolation ProbleIIl

so that both (2.20) and (2.21) hold. At this point it is convenient to pass from the unbounded upper haIf plane to the compact disk. Then Ur,g,N is a rational function in the disk and by Green's Theorem (or the residue theorem) we have for each J E Hl(]]}),

1. -2 7r~

iTf J(z)urg' ,N(z)dz

1 () ~l

-1. 27r~

J(z) ~U1 g N(z)dZ Adz

J[JI

uZ

J(z)d/1-r.g N(Z)

Since the restrictions of the function:> U r g N to the cir de lie in a fixed norm closed ball in L=('JI') = Ll ('JI')*, Corollary A.45' (p. 167) in Appendix A shows that there is a sequence of these restrictions that converges weak star to a bounded function U on 'JI'. We may assume (2.20) holds ab well for the sequence, and in fact the stronger assumption that IIw Nil - t 0 From this we then have that IIwrg.Nllcarleson - t 0 and so by Corollary C 9 (p 187), y

:

We now claim that the theory of the Poisson integral implies

(223)

To see this we use Corollary C 8 to write! = gh with g, h E H2 and Iigli H 2 IIhIl H2 • Now if zJ is the center of K J , then for Z E K J ,

1I!IIHl

2.4. Peter .Jones' COnstructive p~f' of' Interpolation

29

since the K j have hyperbolic diameter about c. Of course we also have !g(z)\ IlPg(z)! ~ IP !g! (z), with similar estimates for h. From these estimates we have

lin

-1, 9(Z)h(Z)dI!r,t;,N(Z)!

g(z)h(z)dp,(z) =

L (

lK

J

~L

(

=

{[g(z) - g(Zj)]h(z) + g(Zj)[h(z) - h(Zj)J}dp,(z) j

{\g(z) - g(Zj)! Ih(z)!

+ !g(Zj)llh(z) -

1K3 ~ Cc ~ [j IP Igl (z)1P Ihl (z)dp,(z)

h(Zj)l}dp,(z)

1

1

~ Cc ( l (IP Iglfdp,) ~

1

( l (IP !hJ) 2dp,)

2

Now we use the nontangential maximal function NG(ei6 ) == sUPzEre !G(z)l, where r6 i& the convex hull of {ei6 } U ~1Dl, to obtain that (see the proof that 4 implies 3 of Theorem 2.4 above)

l (IP Igl) 2dtt

~C

i

~C

NIP Igl (e i6 )2dfJ

i Ig!2

=

C

IIgll~2'

and similiarly for h Thill>

IlI(Z)dIL(Z) -

10 I(Z)dI!.,t;,N(Z) \ ~ Cc IIgll H2 IIhllH2 = CC II/11H1 ,

and we now obtain (2 23) as c ~ 0 Altogether then we have that u

tz

~

{ I(z)u(z)dz =

lT

2n

Now suppose that dJL(z) below we see that

= JL in the sense of Green's Theorem: 11"

I(z)dtt(z).

~ ( G«() d( 1\ d(

in (-

211"~ satisfies F E Cl (1Dl) n CO(jj) and solves tzF 211"~

(

lID

= G(z)dxdy is a Carleson measure. Then from (3.21) F(z) =

~

~

iT( l(e

i6 )[F(ei6 ) -

Z

= tt Thus by Green's Theorem

u(e i6 )]d(ei6 )

~[/(z)F(z)]dZ 1\ dz - ~11" io( I(z)dp,(z) lD uZ 1 -2 l(z)G(z)az 1\ dz - ~ I(z)dtt(z) = 0, n 11" n

1. { = -2 1I"t

=

·1

11"~

1

for all 1 E Hl(IDl), so that F(e i6 ) - u(e iO ) = h(ei6 ) on l' where h E HOO(IDl) Thus if we extend u to the disk by

u(z) == F(z) - h(z),

z E 1Dl,

then u E Cl (1Dl) n c°(ij) is a solution to (2.18) that is bounded on the circle l' with supremum bound controlled by the Carleson measure norm of tt.

30

2. The Interpolation problem

2.4.2 A continuous version. Jones also constructed in [22] a bounded (on the boundary) solution to (2.18) directly from a kernel modelled on the linear operator of interpolation in (2.19). Here are the details in the setting of the complex upper half plane C+ = lR.~. Define

K o(u,Z,()=2i( 7r

z-

~~(Z -

()exp{j {

Jlmw:$Im(

for 17 a positive Carleson measure on C+ and z, ( Carleson measure on C+, set

(z=i_+(~_)du(W)}, w w E

C+, and for 11 a complex

Theorem 2.10 Let 11 be a complex Carleson measure on C+. Then w~th 80(11) as above we have 1. 8 0 (11) E Lioc(C+) 2. :Z80(11) = 11 in the sense of distributions

3.

'

f fc+ IKo( II~WL, x, () Id 1111 «) ~ C 111111 Car for all x

E lR.

OC+, so

~ C 111111 Car· Proof: We first prove assertion 3. We have

1180 (11) lIu">(IR) Re

(_~_. ) = (-w

Re (i(, -W») = Im( + Imw < l(-wl 2

if Imw ~ Im( It follows that if 17

Rei (

= ~I~I ' ~

_ i du(w)

Jlmw:$Im( ( -

1(-wI 2

w

<

then lIulicar

I(

Jlm"':$Im(

<

-

/I

=

2Im(

1(-wI 2

1 and

I«2Im( Idu(w) - w)2

(~~mw~2I1Hl lIu/icar ~ Co,

since the functions f«w) = &~:f2 E HI with norm bounded independent of (. Also, for x E lR. we have Re (

-i_) = (-i(X - ~») = _ Ix - wi Ix - wi Re

X -

Irnw 2'

w

which yields the estimate

I 1+ IK

o(u,x,()ldll1l

<

«)

~I Jc+ { IIIm~:2ecoexp{_1 x-

(

I

Im~12du(W)} /lI1I1Cardu«(),

Jlmw:$Imc, X - w

and assertion 3 follows from the inequality

I Jc+{ Ix - (I

IIm(1 2 exp{-j ( Jlmw:$Im(

Im~2dU(W)}dU«) ~ C,

Ix - wi

(2.24)

valid for any positive sigma finite measure 17 on C+. This is the continuous analogue of the discrete inequality (2 17), and is also proved by showing that if 17 = 2::=1 Cl.j~(j is a finite sum of weighted Dirac measures, then the left side of

2.4. Peter .Jon_' constructive proof of interpolation

31

(2.24) is a lower Riemann sum for the integral Jooo e-tdt - then take limits. Indeed, if Im(j

~ Im(j+l

and {3j

= aj !!Im'i!6,

then the left side of (2.24) is

X-'j

Assertion 1 follows from the estimates above and it remains to prove 2 We begin by using ~ (z~') = 7r8, in the distribution sense (see (3.22) below and note that d( A d( = -2id1;d'1) Then for any well-behaved kernel L(z, () that is analytic in the first variable, we have :ZL(z, () = 0 and so

! {j L+

(z

~ () L(z, ()d/L«()}

=

j L+ (! (z ~ () )L(z, ()d/L«() 7r

j L+ 8,(z)L(z, (}d/L«()

7r L(z, z)d/L(z),

in the di&tribution sense. If we write Ko(CT, z, () then we obtain ()

OzSO(/L)(z)

= (z~() LO(CT, z, () and CT = III}r~ar'

=

7rLO(CT, z, z}d/L(z)

=

2i Imz 7 r - -_)d/L(z) ( 7r Z - Z

= d/L(z)

For convenien('e we state the analogue of Theorem 2.10 for the unit disk Jl] obtained from Theorem 2 10 using the conformal map z --t i~ from Jl] onto C+. A& in [52] define 1 - 1(1 exp{j f (_ 1 + ~z 7r (z - ()(1 - (z) Jlwl~1(1 1 - wz

K(CT, z, () == 2i

2

for CT a positive Carleson meabure on measure on Jl], set

Jl]

+ 1 + ~( )dCT(w)}, 1 - w(

and z, ( E D, and for /L a complex Carleson

z E

Jl].

The and-logue of Theorem 2 10 is: Theorem 2.11 Let /L be a complex Carleson measure on Jl]. Then with S(/L)

as above we have 1. S(/L) E Ltoc(Jl])· 2. :ZS(/L) = IL in the sense of distributions.

3.

I In lK(IIIJr~ar,x,()1 dl/L! «) ~ C II/Lllcar for all x E 1[' = em, so IIS(/L} II Loo(T) ~ C 1I/Ll!car·

32

2. The Interpolation problem

2.5 Other interpolation probleInS

In a revolutionary paper in 1994, D. Marshall and C Sundberg [24] used Hilbert space methods (and independently C. Bishop [10] used different techniques) to characterize interpolating sequences for the Dirichlet space B 2 (1!}) and its multiplier algebra M B2 (D) (note the connection HOC(I!}) = MH2(D» by the condition f3(Zi,O):S; Cf3(Zi,Z) for i -=1=] and

f (1 + j=l

log

1 1 -lzJI

2) -I 8

Zi

if> a

(225)

B2(1!})-Carle~n measure,

where f3 is the Bergman metric', and a positive Borel measure J.L is a B 2 (1!})-Carleson measure if the embedding B 2(1!}) C £2(dJ.L) holds·

J

2

I/(Z) I dJ.L(z):S; C

11/118.2 2 (0)

A crucial part of their argument used the Nevanlinna-Pick property of B.2(I!}) an important consequence of this property for any Hilbert space X of analytic functions on the disk, is that X then has the same interpolating bequenCef> as its multiplier algebra Mx (Theorem 56 (p 94». The above two conditions are analogous to the separation and Carleson conditions that arose in Carleson's Interpolation Theorem above. In fact, with the definition

_I Zi -

d(Zi' Zj) =

Zj

1

1 - ZJzi

,

and using the identity (26) in the form d(Zi,ZJ)2 + l(kz"kZ j

)1

2

=

(226)

1,

the separation condition (2.7) above can be rewritten for some c> 0 and all i -=1=], and the Carleson condition 4 above can be rewritten oc 1 ---28zj is a Carleson measure for H2,

L

J=I

JJkzJ

where kz«) = I~
£. =

VI -lzl.2 kz

=

II~~II

The function d is a metric on I!} called the PbeudohyperboliC' metric, and can be generalized to the Hilbert function spaces treated in Section 5 Because of the identity (226), d(Zi' zJ) can be thought of as the sine of the angle {}i1 between kz. and kZj This interpretation leads to the following cute proof that d is a metric (Lemma 9.9 in [1]) From geometry we have {}il :s; {}ij + {}jl If the right side is at most ~,then otherwise, we have

33

2.5. Other interpolation problems

Finally, there is a formula relating the Bergman and pseudohyperbolic metrics (Proposition 1.21 in [53]) which we state without proof 1 1 + d(z,w) j3(z, w) = 2" log 1 _ d(z, w) The two conditions in (225) can be rewritten in exactly the above form with only the natural changes; the k' b must now be normalized reproducing kernels for the Dirichlet space, and the mE'abure must be a Dirichlet space Carleson measure 2.5.1 Further results. In 2002 in [11], B Boo has extended the above theorem to all 1 < p < 00 by a long and clever construction involving Carleson measures, that. was in turn ba.bed on an earlier construction in Marshall and Sundberg [24] (see also thE' analogous construction on treeb in Section 6 of [4]), together with, in Boe's words, a "curioUb lemma" on unconditional basic sequences {/j} J=1 of positive functions in a Lebesgue space Lq(d/l) 00

~ C II~~~ lajlilll q

Lq(d/L)

)=1

-

Lq(dJ.L)

The sequence {fJ }~1 is an unconditional basic sequence if 00

00

"'£bd) ::; C "'£aj/J )=1

)=1

whenever Ib1 1 ::; laj I Recently, Arcozzi, Blasi and Pau [3] have solved the interpolation problem for all of the Besov-Sobolev bPaces B;(ID» in the disk, 0 ::; u < ~,1 < p < 00, in ternlb of separation a.nd Carleson embedding conditions by using the constructive approach of Peter Jones. \VhE'n we defined "interpolating sequence" for the Hilbert space H2 we required that the rest.riction map R take H2 both into and onto ,e2(Z, /lz) In fact we showed in Theorem 24 that if R is onto (condition 2) it must also be into (condition 3). However for the clabsical Dirichlet space t.he map can be onto without being into Hence one can ask for a characterization of those maps for which R is onto The question remains open. partial results are in [10], [11] and [6]. \Ve will extend horne of th~e resultb to higher dimensions in later sections We characterize interpolating sequences fOI X and its multiplier space Mx in terms of separation and Carleson embeddings when X = B~(Bn) with u E [O,~) For this we give a proof in Theorem 5 31 (p 112) using Hilbert space techniques including the Nevanlinna-Pick property In [5], Arcozzi, Rochberg and the author characterizE' interpolating sequences for the Besov spaces B~(Bn) and their multiplier algebras in higher dimension when p E (1,2+ n~l) U (2n, oo). This uses an extension of Boo's method in the disk, the characterization of Carleson measures in Theorem 6.9 (p. 141), and a "holomorphic" version of the Bergman tree Tn in which an appropriate analogue of Boo's lemma holds. We refer t.he reader to [5] for the detailb. The interpolation problem for n > 1 remains unsolved for all other Besov-Sobolev spaces and their multiplier algebras.

CHAPTER 3

The Corona Problem The continuous functions G(K) on a compact set K provide the prototypical example of a commutative Banach algebra Gelfand's theory shows that an arbitrary Banach algebra X has a natural homomorphism~: X -+ G(M) with kernel radX (the intersection of all maximal ideals) into G(M) where M is the maximal ideal space of X,

M = {

3.1 Commutative Banach algebras We collect some simple results and examples in Gelfand's theory of commutative Banach algebras as presented in [381. Recall that a commutative Banach algebra is a (complex) Banach space X with a commutative associative mUltiplication with unit e that distributes over addition and satisfies

!!xyl! ~ Ilxll !Iyl!,

x,y

E X

A complex homomorphism h is an algebra homomorphism from X to
Theorem 3.1 Let X be a commutative Banach algebra, and let M be the set of all non-zero complex homomorphisms of X. Then 35

36

3. The Corona Problem

1 If IIxli < 1 then e - x is invertible and !h(x)! < 1 for all hEM. Moreover,

II II (e-x) -1 -e-x::;

IIxll2 1-lIxli

2. Suppose x is invertible and h E B(x, 21Ix1- 1 11)

II (x + h)-l -

Then x

(3.1)

+ h is invertible and

x-I + x-1hx-11l ::; 21Ix-lhI12I1x-lll ::; 21Ix-1113I1hIl2.

(3.2)

3 The spectrum O'(x) is compact, contained zn p. E C I,X,!::; 1I.e1l}. and nonempty for all x EX. 4. Every maximal ideal M is the kernel of some hEM and vice verba 5. An element x E X is invertible if and orlly if hex) -=J- 0 for every hEM if and only if x lies in no proper ideal 6. ,X, E O'(x) if and only if hex) = ,X, for some hEM Remark 3.2 The algebra X need not be ('ommutatiYe for the first three assertions, and for these we will give.> proofs that work in the nonC'Ommutative ('a~e as well

Proof' (1) Since II x" II ::; IIxll n and IIxll < 1, the el('ments Sn = e + x + form a Cauchy sequence in X. and hence converge to some [, EX. Then sn(e - x)

. + x1t

= e - x n +1 = (e - x)sn

and taking the limit as n --+ 00 yield5 s( e - x) = e = (e - x)s Thu~ e - x i~ invertible and Sn - e - x = x 2 (1 + x + .. + x n - 2 ) yields (3.1) Finally, if I,X,I ~ 1, then y = e-'x' -IX is invertible, so 1 = h(e) = h(y)h(y-I) implies that I-,X, -1 hex) = h(e - .x,-IX) -=J- O. (2) Since x + h = x(e + x-Ih) and Ilx-Ih\\ < ~, pa.lt (1) shows that x + h i~ invertible. Then (3.1) and the identity (x + h)-I - X-I + x-Ihx- I

{(e + x-1h)-1 - e + x-1h}x- 1

=

(multiply on the left by x and use e + .e-Ih = x-lex + h» imply (32) (3) By part (1) ,X,e - x = 'x'(e - ,X,-lx) is invertible if I,X,I > IIxll, so O'(x) c {'x' E C : I,X,I ::; IIxll}. By part (2) the set G of invertible elements in X is open, and hence so i~ p. E C . 'x'e - x E G}, and the complement O'(x) is closed, thm; compact. Now suppose in older to derive a ('ontradiction that O'(x) is empty. Then fx' (,X,) == ('x'e - X)-I, x*) is an entire function since if we write y = 'x'e - x and h = (M - 'x')e. then by part (2),

II(Me - x)-l - (,X,e - X)-I + (M - 'x')('x'e - x)-211 = \\(y+h)-I_y-l +y-1hy-111

~ 2 Ily- 1 1l3 IIhll 2 and so

=

2 l\y- 1 1l'3 1M _ ,X,12 .

37

3.1. COlDlDutative Banach ;'lcebras

shows that I~*P..) = «(Ae - X)-2,X*) exists for all A E C. Now Ix* vanishes at infinity and hence Ix. == 0 by Liouville's Theorem Thus (x-l,x*) = -Ix· (0) = 0 for all x* E X* which implies X-I = 0, the desired contradiction. (4) If hEM, then ker h is a maximal ideal since it has codimension 1. Conversely if M is a maximal ideal, then Me contains the open ball B(e, 1) by part (1), and hence so does Ar. Thus M J M is also a proper ideal, and by maximality M = M. Thus M is closed and by straightforward calculation the quotient algebra XI M is a commutative Banach algebra. Choose x E X \ M and let J = {ax

+ y' a E X,y EM}

be the ideal geneIated by x and M Then J ~ M and by maximality J = X. Let e = ax + y with a E X and y EM. Let 7f : X -> X \ M be the quotient map and note that 7f(e) = 7f(a)7f(x) shows that every nonzero element in the quotient Banach algebra Y == X \ M is invertible. Now for any y E Y and Al =I- A2, at most one of thE' element& >'1 e - y and >'2e - y is zero, hence at least one of them is invertible Since a(y) =I- ¢ by part (3), it follows that a(y) consists of exactly one point, say >.(y) E C Since >.(y)e - y is not invertible, it must vanish and y = >.(y)e. Thus the map >. . Y -> C is an i&ometric isomorphism, and if we define h = A 0 7r, then hEM and M = kerh. (5) If x is invertible and hEM, then 1 = h(e) = hex-Ix) = h(x- 1 )h(x) shows that hex) =I- 0 If x i& not invertible, then the ideal {ax. a E X} generated by x is proper, hence contained in a maximal ideal M by a standard argument using the Ham,dorff maximality principle (we seem to need the axiom of choice here even if thE' algebra X is separable). By part (4) M is the kernel of bOrne hEM. Thus hex) = 0 By (4) we have that h(x) = 0 for some hEM if and only if x E M for some maximal ideal. (6) By part (5) >.e - x is invertible if and only if>. - h(x) = h(>.e - x) =I- 0 for all hEM Now equip M with the weakest topology that makes the family of functions

X=

{x: x

E

X}

continuoill> on M Here x(h) = hex) for hEM With this topology on M we define the Gelfand transform A . X --> G(M) by Ax = x E X.

x,

Theorem 3.3 Let M be the maximal ideal space of a commutative Banach algebra X Then 1 The Gelfand transform A . X -> G(M) is an algebra homomorphism from X onto a subalgebra X of C(M) and the kernel of the Gelfand transform is radX, the intersection of all maximal ideals in X 2. The range ofx is the spectrum a(x) for each x E X, and the Gelfand transform is continuous with norm 1 IIxIlC(M) ~ IIxll, x EX. 3. j\lf is a compact Hausdorff space Proof· (1) Chasing definitions shows that the Gelfand transform is an algebra homomorphism. The kernel of A consists of those x E X such that 0 = x(h) = h(x) for all hEM, i.e. ker A = nhEM ker h. By part (4) of Theorem 3.1, we obtain kerA = radX. (2) Now A is in the range of x if and only if A = x(h) = h(x) for some hEM. By part (6) of Theorem 3.1, this ha.ppens if and only if>. E a(x). From part (3)

38

3. The Corona problem

of Theorem 3.1 we conclude that IIxlic(M) ~ IIxll, and so the Gelfand transform is bounded with norm 1. (3) Let C be the norm closed unit ball in the dual space X*. Then C is weak* compact by the Banach-Alaoglu Theorem. By part (1) of Theorem 3.1, M C C. Since the Gelfand topology on M is the restriction of the weak* topology on X*, it suffices to show that M is a weak* closed subset of X*. But this follows from the standard argument demonstrating inheritance in weak topologieb: suppose that Ao E M and consider the weak* neighbourhood

W = {A E X* : IAzi - AOZil < e, 1 ~ i ~ 4} where Zl = e, Z2 = X, Z3 = y, Z4 an hEM and it follows that

=

xy and e

> 0 Here x, y

E

X

Now W contains

11 - Aoel < e, IAo(xy) - Ao(x)Ao(Y)1 < (1 + IIxli + IAo(Y)I)e upon using the triangle inequality to compare Ao and hand ru,ing h(e)

= 1 and h(xy) = h(x)h(y) 3 1.0 1 Examples Now we give some examples of Banach algebras X, and ('ompute their maximal ideal spaces M and Gelfand transforms

1. First let K be a compact Hausdorff space and let X = G(K) be the Bana('h algebra of continuoub funetions on K with pointwise multiplication and the supremum norm In this case the underlying space K is homemorphk to the maximal ideal space M under the map tP K ----+ M where tP(p)(f) = f(p), P E K, is the point evaluation homomorphism at p. Since K is Hausdorff, G(K) separates points on K by Urysohn's lemma, and tP b a ('ontinuous embedding \Ve claim it b also onto. Indeed, if not, then there is a maximal ideal M that b not contained in any kertP(p), i.e for every p E K, there is f E M such that f(p) =I- 0 By compactne&s of K there are finitely many functions fro . ,IN in M bu('h that N

g(P)

=

L

fJ(p)/j(p) > 0,

pEK

(3 3)

)=1

But then gEM and g b invertible in G(K), a contradiction. If we identify K and M via the map tP, then the Gelfand transform i& just the identity map from X = G(K) to itself G(M). Indeed, for f E X and p E K chasing definitions yields j(tP(p» = tP(p)(f) = f(p) Moreover. the Gelfand topology T on M = K coincides with the original topology "/ on K by rigidity of compact Hausdorff topologies. Indeed, T is Hausdorff and 1 ('ompact. Sin('e T C "/ by definition, both topologies are compaet and Hausdorff, and we must then have T = "/ (Proof: the identity map Jd takes (K, ,,/) ('ontinuously one-to-one onto (K, T), so if G E ,,/, then GC is compact in (K, ,,/), Jd(GC) is compaet in (K, T), hence G' is closed in (K,T) and GET) 2. Let][»n = ][» X •• x][» be the polydisk in en and let A(][»n) be the polydisk algebra consisting of those analytic functions in ][»n that admit a continuous extension to the closure ][»n. Then X = A(][»n) is a Banach subalgebra of C(][»n). We showed in the first example that the maximal ideal space of c(][»n) can be identified with ][»n, and we show here that the same is true of X = A(][»n) Again ][»n is continuously embedded in the maximal ideal space M of A(][»n) by the point evaluation map tP: ][»n ----+ M where tP(z)(f) = fez), Z E ][»n. To see that tP is onto,

3.1. Commutative Banach alpbr_

39

choose hEM. Let 9j(z) = Zj be the coordinate functions for 1 :.:; J :.:; n. Since !!gjl! :.:; 1, we have w = (h(gd, ... ,h(gn» E JI)n by part (1) of Theorem 3.1. Now polynomials P(z) are in A(JI)n) and so we have pew) = h(P) for every polynomial P since h is a homomorphism. Furthermore the polynomials are dense in A(JI}n) (if I(z) E A(JI}n) and r < 1, then Ir(z} = I(rz) is given by an absolutely and uniformly convergent multiple power series in JI)n), and it follows that I(w) = h(f) for all I E A(Dn) by the continuity of h. Thus h =
11,· ·,/N

E A(Dn) satisfy

there are gl,. ,gN E A(Dn} such that

I:f=l !1;(Z)!2 >

I:f=l I;(z)gj(z)

= 1

for

0 lor z E Z

JI)n,

then

E JI)n.

If the claim fails, then the ideal J = {I:~1 hgJ : gj E A(JI)n)} generated by {h, ... , IN} is contained in a maximal ideal M, hence by part (4) of Theorem 3.1, J is annihilated by some hEM. By what we proved above h =
3. Let X = HOO(JI) be the Banach algebra of bounded analytic functions in the unit disk JI). Then D is embedded continuously in the maximal ideal space M of HOO(D) by the point evaluation map
N = {A EM· IA(IPj) - h(IPj)! < E, 1 :.:; j :.:; N}

(34)

contains a point evalutioll A = 0 and IP1, ... ,IPN E Hoo (D) (recall from above that the Gelfand topology OIl M is the restriction of the weak* topology on X*) We now claim that
11, .. , fN

E

HOO(D) and 8> 0 satisfying

N

L

!IJ(z)12 ;::: 8 > 0,

zED,

(3.5)

j=l

there are gI, ... ,gN E HOO(D) such that N

L

h(z)gJ(z) = 1,

z E ID.

(3.6)

j=l

Suppose first that the criterion holds, and in order to derive a contradiction, that 0 and {IPl' ... , IPN} C HOO(D) such that
3. The Corona probleHl

40

the functions h = (J]}) is dense in M, and in order to derive a contradiction, that the criterion fails. We argue as in the proof of the claim in example 2. There are functions {II, .. , IN} c Hoo(J]}) satisfying (3.5) such that the ideal J = {L~l hgj gj E Hoo(J]})} generated by {II,· ,IN} is contained in a maximal ideal M By part (4) of Theorem 3.1, J is annihilated by some hEM. Since cf>(J]}) is dense in M, there is for each n ~ 1 a point Wn E J]} such that cf>(w n ) EN wherE' N is as in (3.4) with E = ~ and
i~(~~;'):)z is easily seen to satisfy limz_eill IB(z)1 = 1 n=l for all () E [-1r, 1r) \ {Ole Then with Ba:(z) = B(e-iOtz), thE' uncountable set of functions S = {B(e- i a:z )}a:E[_1r.1r) lies in the closed unit ball of Hoc, yet II Ba: - B.6l1oo = 1 for -1r::::: a < f3 < 1r since limr_l IB,8(re i a:) = 1, while Bc.(reiCt ) = 0 for r in the set {(I - 2-n)}~=1 which clusters at 1. Blaschke product B(z)

=

IT

I

3.2 Wolff's proof of Carleson's Corona Theorem

We follow the development in [20]. It suffices to establish Criterion 35. So let

{J) }f=l be corona data as in (2.1), i.e. a finite set offunctions in HOO(IDl) satisfying J

L

II)(z)1 2 ~ 82 > 0,

z

E

(3.7)

IDl

j=l

We may assume the h are analytic in a neighbourhood of the closed disk and obtain corona solutions gJ with bounds independent of the neighbourhood Then a normal families argument will complete the proof. In an initial attempt to construct corona solutions {gj };=1 as in (2.2), i e J

f· g

==

L IJ (z)gJ (z) = 1,

(3.8)

z E 1Dl,

)=1

where fez) = (

JI(z) ) .

and g(z)

h(z)

=

( gl(Z) )

:

,

gAz)

we write -

J

I - fez) ·f(z) _ ~ . z

--

h(z)

- f(z)· fez) - f;:;.IJ( ) fez) f(z)'

so that J

fez) .
=

L j=l

IJ(z)
(3.9)

41

3.2. Wolff's proof of Carleson'. Corona TheoreD1

with !pJ'(z) = I (t;W for 1 < j :::; J Now!pj E Coo(lI))) n LOO(II))) with II zW+ +1!J(z)1 2 LOO bound at most C II f ll oo ' but in general the 'Pj will not be analytic In order to construct g = (gJ);=l from V' = (!PJ)f=l where the gj are bounded by C(8, II f lloo), analytic and satisfy (3.8), we define g(z)

= V'(z) + A(z)f(z),

(310)

for an appropriate matrix function A (z) = [ai) (z )1t,j = 1· Indeed, if A(z) is antisymmetric, then the quadratic form v· A(z)v v = -A(z)v. v is identically zero and (38) holds since

=

A(z)'v.

J

~ /jgj = f . g = f V' + f Af = f· IP = 1, j=l

where the last identity is (3.9). To obtain the boundedness and analyticity of the n LOO(II))) with L oo bounds at most C(8,lIfIl00)' solving the equations

g) it buffices to find ai) E Cl(lI)))

1:::; i,j:::; J,

(3.11)

i.e

oA

alP'

all',

Oz

Oz

Oz

-=IP- --IP

Indeed, g is then obviously bounded by C(8,

IIflloo) and

og Oz

since IP . f = 1 by (3.9), and ~ . f = %z(V'. f) = %z1 = 0 since f is analytic. Thus the gj are in HOO(II))) with LOO bounds at most C(8, IIflloo)' and satisfy (3.8) as required. This procedure arises from the Koszul complex (see e.g the appendix to Chapter VIII in [20], and also Subsubsection 5.5 3 (p. 128»), and its application to the corona problem was noticed by Hormander. So it remains to solve (3.11) with aiJ E Cl(l!}) n LOO(l!}) with lIaij!loo :::; C(8, IIfll oo )' and for this it suffices to solve

8b -(z) Oz

=

G(z)

'

zE

11)),

(3.12)

42

3. The Corona problem

with b E C1(D) n LOO(D) and IIbll oo ~ C(~, IIflloo)' where G(z) =
a

-

=

a( &z

7iW

L~=l Ih,(z)1 2

lIW

=

L~=1I!k(z)12

(L J

t(

k=l !k

)

(3.13)

Y;W(L~-l /k(z)fITZ) (L~=l J!k(Z)J2)2 J

)12)2 'L,fk(Z){Jk(Z)fj(z) - ff.(z)fJ(z)}.

Z

k=1

From (313) we have IG(z)J ~ C(~, IIflloo) Jff(Z)J. A crucial consequence of (3 13) was observed by Wolff' since derivatives of f appear there only as conjugate holomorphic functions, we have

(3.14)

tz·

an inequality that in general fails with tz in place of At this point we only know that G E Coo (D), and from Cauchy's inequality if(n)(z)l ~ n! IIflloo R-n with n = 1 we have the trivial bounds JG(z)J ~ C(~, IIflloo)(1-lzj)-1 and JG'(z)1 ~ C(~, IIfllo,,)(1 - Jzj)-2, which show that each of the three functions JG(z)J, (1 -Jzj) JG(Z)J2 and (I-Jzj) JG'(z)J

(315)

is in weak LI(D) with "norm" at most C(~, IIflloo). Now recall that a measure f.L is a Carleson measure for H2(D) if IIfll p (]J),JL) ~ C IIfIlH2(]J)) for all f E H2(D), in which case we define the Carleson norm 1If.Lllcar(H2()[))) to be the least constant C for which the embedding inequality holds

Remark 3.7 If Jf'(z)1 dxdy is a Carleson measure, then Jones' interpolation solution to (2.18) adapted to the disk, or more precisely Theorem 2.11, yields the required bounded solution to (3.12). However, dw(z) = Jf'(z)J dxdy is not in general a Carleson measure for f E Hoo(D). Indeed, there are Blaschke products B(z) with f]J) JB'(z)1 dz = 00 (see Exercise 9 on page 264 of [20] for this and more). Since Iff(z)1 dz is not necessarily even integrable, we see that the first of the functions IG(z)1 in (3.15) is not in L1. However, Wolff's clever observation is that not only are the other two functions (1- Izl) IG(z)J2 and (1 - Izi) JGf(z)1 in (3.15) integrable, but that both of the corresponding measures df.LI

== (1- JzJ2) IG(z)J2 dxdy and df.L2 == (1- JzJ2) JG'(z)J dxdy

are in fact Carleson measures for the Hardy space H2(D). Furthermore we have bounds in terms of the Hoo(D) norm of the corona data f in (3.7),

(3.16) and (3.17)

3.3. The corona d-bar equation

43 .l

To see this we use (1.4 (p. 2», which says that H2(D) = Bi (D) where by (1.1),

11/112 ! B2

(JI)

~

iJl)r1f'(z)1

2 (1-lzI2)dxdy + 1/(0)1 2,

together with Lemma 2.9, which says that the mUltiplier algebra is isometrically isomorphic to Hoo (D) It is an easy consequence of these two facts that

(3.18) MH2(JI)

of H2(lDl)

1

is a Carleson measure for

Bi (lDl) = H2(lDl) with hE HOO(lDl).

IIJ1.hIICar(H2(JI))) :::; C IlhIIHOO(JI), Indeed, for

(3.19)

1 E H2(1DJ) and hE HOO(IDJ), we have h'l = -(hf)' + hI', and so 1

II/lIp(JLh)

<

(10 dZ) (1(1 - Iz12) l(hf)'(z)1 2 dZ) + (1(1 -lzI2) Ihf'(z)1 2 dZ)

<

II hili

=

(1-lzI2) Ih'(z)/(z)1 2

"2

1

1

2

2

Bl (D) + II h llw>O(D) 11/11 B? (D)

< (C I hll MH2(1l) + II h IlHoo(JI)) II/I1 H 2(JI). If we now take h = fJ, 1 :::; j :::; J, and use that
Now we solve the d-bar equation (3.12) using essentially Gamelin's variation on Wolff's argument (see e.g section 2 and exercise 5 of Chapter VIII in [201). This variation doesn't use Carleson's characterization of the embedding H2(1DJ) C 1.

L2(1DJ, J1.), but will use the identification of H2(1DJ) with Bi (1DJ) and the characterization of pointwise multipliers given above. We will assume that G is bounded (which follows from our assumption that the corona data f are analytic in a neighbourhood ofthe closed disk), and construct a solution b E C 1 (1DJ) n C(ll}) to (3.12) whose supremum on the boundary of the disk is bounded by a constant depending only on the Carleson measure norms of ILl and J1.2, which in turn depend only on the HOO(IDJ) norm of the corona data f. Recall that dz A az = d(x + iy) A d(x - iy) = -2idxdy. We begin with the convolution integral F(z) =

~ 27[%

f inr(G«() d( Ad(, z

(3.20)

which is easily seen to satisfy FE C1(IDJ)nC(ll}) if G E Cl(lDJ) is bounded. However, at this point the supremum norm of F on the circle 'Jr depends on that of G, and this key difficulty will be addressed below. We first claim that

of

Oz (z)

= G(z),

zED.

(3.21)

3. The Corona problem

44

Indeed, for 1fJ E C~(D) Green's Theorem yields both

1fJ(z) =

{f ~I f Ii ~«()

1fJ«() d( -

lim _1

A;-zl=e ( - z

e ..... O 21ri

1. -2 1rZ

21rz

1fJ«() z

d(}

(322)

~( 1fJ«() )d( A d(

-lim

e-+O

f

i,t;- zl=l ( -

iD\B(O,e)

o( ( - z

-t"-d(Ad(,

11> '> -

Z

and

- / /

O~1fJ) az A dz

- / F1fJdz = 0 Using (3.20) with the above and Fubini's Theorem we thus obtain

~~ (z)1fJ(z)dz A az

/ /

- / / F(Z/';: (z)dz

A

_//{_1 / ill>f ( -

az

G«()d(Adz'}O::(Z)dzAaz

21rZ

Z

- / iDf G«(){~// 21rZ =

/

uZ

~(Z)dzAaz}d(Ad(

(- z

/ G( ()1fJ( ()d( A de·

Since this holds for all1fJ E C~(D) we conclude that ~f = G in D, i e (3.21). Now every solution b E C 1 (D) n cOi) of (3 12) has the form b = F + h where h is analytic (~; = 0) and lies in the disk algebra A (D) = H= (D) n C (D). The minimal norm of such solutions b to (3.12) is therefore given by inf

hEA(D)

lIF + hll£'x>(1r) = sup{

/1

211"

0

I

dO Fk-. k 21r

E

HJ(D), Ilklll S; 1}.

(323)

Indeed, for any Banach space X with closed subspace Y, the Hahn-Banach Theorem gives the isometric embedding (XjY)* ~ y.L (Proposition A.41 (p. 166», i.e for x E X and x* E X*, inf{lIx + yll . y E Y}

= sup {I(x, k)1 . k E y.L, IIkll

S; I}.

(324)

Now take X = Cpr), Y = A(T) and (I,p,) = f~1I" J(O)dp,(B) in (3.24) The F. and M. Riesz Theorem C.lI in Appendix C shows that y.L = HJ(T), which yields (3.23).

We may suppose that k E HJ(RJDl) for some R> 1 From (3.21) we have

L.(Fk) =

4~~(Fk) = 4~ (OF k) = 4~(Gk), OZ Oz oz Oz OZ

and Green's Theorem in the form

h

(uvn - unv) =

I1

(u L. v - (L.u)v),

45

3.4. Other corona probleDls

with u = Fk and v = log~, now yields using with 6v = 80 and k(O) = 0,

l

k

o

F(eiO)k(e iO )_ • 21r

IJl ~ Jl

-

21r

Vn

= 1 and v = 0 on 'll', together

6(F(z)k(z)) log -I 1 Idxdy Z

ITJ)

k'(z)G(z) log

2J rk(z) 8z (z)

+; I

8G

JITJ)

1~ldXdY 1 log ~dxdy

+ II.

We estimate integral I using the factorization k = Ig with I, 9 E H2(]!))) and 1 (Corollary C.8 (p. 187) in Appendix C), along with the CauchySchwarz inequality as follows

11/1I2,lIg1l2 ::;

III

< 0 < 0

J

llUg'

+ g/')(z)IIG(z)1 (1 -

Izl2)dxdy

(J ll/(z)12IG(z)12 IzI2)dXdy) (J llg'(z)1 2 Iz I2)dXdY) (1 -

1

"2

1

X

(1 -

"2

+ a similar term corresponding to 9 /' , < 0 IItLIIICar(H2(ITJ»)) 1I/I1211g112 ::; 0(0, IIfIlHOO(ITJ»))' by (3.16) and the case m = 1 of (3.18). To estimate I I we proceed as follows:

IIII <

0

< 0 x

J

ll/(Z)9(z)l\ ~~ (z)\ (l-lzI 2 )dxdy

(J ll/(z)1 2 \ ~~ (z)\ -l z I2)dXdY) ~ (J llg(z)1 2 \ ~~ (z)\ (1 - Iz I2)dXdY) ~ (1

< 0 IltL211~ar(H2(ITJ»)) 11/112 Ilglb ::; 0(8, IlfIIHoo(D»)' by (3.17). Thus from (3.23) we conclude that the minimal LOO('ll') norm of solutions bE C 1 (]])) n C(IDl) to (3.12) is at most a constant O(IJfIlHoo(D») depending only on

[lfIIHOO(Il» Thus for corona data f E HOC(]])) and R < 1, there exist solutions aij E Ol(]!))) n C(IDl) to (3.11) with fR in place of f, and where [[aijlloc ::; 0 independent of R. From (3.10) we see that there is gR E HOO(]])) solving (3.9) with fR in place of f and with IlgRIIHoc(ll» ::; C(8, IIfIIHoo(][j))). Now a normal families argument shows that there are bounded analytic solutions gj to (3.9). This completes the proof of Carleson's Corona Theorem. 3.4 Other corona problems

We first present proofs of the corona theorems for the algebras HOC (]!))) and MV(][j) respectively, and then briefly discuss further results.

n D(]]))

46

3. The Corona Problem

3.4.1 The algebra HOO(D) n V(D). In [28] Nicolau has proved the corona theorem for the Banach algebra H oo (D) n V(I!») consisting of the bounded functions in the Dirichlet space V(]J}) = {f E H(]J}) : JIf) 1f'(z)1 2 dxdy < co}: if I,g E HOO(]J}) n D(]J}) , then IIlgII H= ~ IIIIIHoo IlgliHoo and

IIlgII~(If)

1

1f'(z)g(z)1 2dxdy +

=

l

lf (z)g'(z), 2dxdy

~ IIfll~ IIgII~oo + IIfll~oo IIgllv, so that HOO(]J) n D(]J) is an algebra. Completeness is obvious. We now reproduce Nicolau's proof in [28] with a simplification due to Xiao [52]. Let {h };=1 be a finite set of functions in HOO(]J) n V(]J) satisfying J

L

Ih(z)1 2 ~ 82 > 0,

Z

E]J).

j=l The Gelfand theory reduces the proof to constructing {gj};=l in HOO(I!») n D(]J) satisfying J

Lh(z)gJ(z) = 1,

z E]J).

j=l

Let W 1 ,2(]J) as in Definition B 3 (p. 174) in Appendix B be the Hilbert space of functions f in the disk with locally square integrable weak derivatives normed by

IIfll~,

2(1f)

=

1IV'

f(z)12 dxdy.

See Theorem BA (p. 174) in Appendix B for the completeness of W 1,2(]J). As in the proof of Carleson's Corona Theorem for HOO(]J), it suffices to solve the d-bar equation (3.12), namely {}b m(z) = G(z), Z E]J), but this time with bE C 1 (]J)nC(D)nWl,2(]J), and most importantly, with control of both the L OO (,][,) and W 1 ,2(]J) norms of b in terms of the HOO(]J) and V(]J) norms of the corona data f = {fJ};= 1· Recall that G (z) = 'Pi (Z) 'P j (z) for any fixed i, J where 'Pi(Z) =

Ih(z)12~lfJ(Z)12

{}

tz

and

1

J

m'P)(z) = (E:=1Ifk(Z)1 2)2

~fk(Z){fk(Z)f;(Z) -

f;'(z)h(z)},

for 1 ~ i,j ~ J. In the present situation, unlike in the proof of Carleson's Corona Theorem, the measure dJ.L(z) = IG(z)ldxdy is actually an H 2 -Carleson measure:

J.L(T(I»

=

r

IG(z)1 dxdy

iT(J)

< C

(r

iT(!)

~C

r

If'(z)1 dxdy

iT(J)

If'(z)1 2 dXdY )

~ IT(I)I! = C IIfliv III·

47

3.4. Other corona problems

Thus from the previous subsection (or Remark 3.7 (p. 42», we see that with

F(z) = -1. 27rt

l

G«()

-

- - d ( Ad(

D (- Z

as in (3.20), then tzF = G and inf

hEA(D)

IIF + hll V", (T)

:::;

C IIf!!Hoo .

Thus there is a solution F + h with IIF + hIlLoo(T) :::; C(l\fll v + IIfllclO). We also have a good estimate for the W 1,2(lJ) norm of F using that tzF = G and {) 1 [G«() -

in «( _z)2d( A d( =

{)z F = 27ri imply

BG(z)

tz F is the Beurling transform BG of G = tzF given by

11-

BG(z) = -2. 7rZ

ID

K«(,z)G«()d( Ad(,

where the kernel K«(,z) = «'!Z)2 is a singular integral on C the theory of singular integrals in [46] we then have

IIFII~Tl 2(ID)

=

klV

= ]R2.

Indeed, from

F(z)12 dxdy

ll!F(Z)r dxdy+ l\:zF(Z)r dxdy

10 IG(z)12 dxdy + 10 IBG(z}12 dxdy and

10 IBG(z)1 2 dxdy

< C

k IG(z}12 dxdy

< C(lIflloo)

L

Jf'(z)1 2 dxdy

=

C(l!fll oo ) IIfll~,

by the boundedness of the Beurling transform B on L2(lJ)}. Remark 3.8 The kernel K«(, z) = the standard kernel estimates

«.!Z)2

of the Beurling transform B satisfies

IK«(, z)1 <

1

IV<,zK«(, z)1 < The cancellation requirements are embodied in the formal equalities Bl(z) B*l(z)

IJ[d(Ad( iD Z)2

27ri

«( -

=

1 XC\D(z) z2'

=

XC\D(Z)z2'

1 J[d(Ad( 27ri

10 «( _z)2

1

3. The Corona problem

48

which in particular establish the hypotheses of the T1 theorem (see e.g. Stein's book [46]). To calculate B1(z) let G == 1 and compute that 1

F(z) = -ZXIli(Z) - -Xc\Ili(z), Z

so that B1 = -/zF = XCW(z):z%: if ( = sei4> and z = reiO, then a change to polar coordinates and the residue theorem formally yield

F(z)

_.!. 1r

{I {21r

io io

-i e - io 1r

'''' 1 '0 d -;-(-;-,A.:-:-)ie 2,,------=r-',-,... d
l {1 i

0

ie-io {I {{

io Ie w

1r

ie-io 1r

{I{ {(i+

io Ie

'l

i _e-'o 1r

2dw r } ds - sW

w

min {r.I}{

0

-;:r)dW}dS w -

s

S} ds = _e-',o min{r, -}. 1

21rir

r

However, instead of using the singular Beurling transform B to estimate the L2(J[))) norm of tzF, we can instead use a clever device of Xiao [52] that moves the singularity outside the disk. Let
t::·A

F(z)

= =

For z E '][' we have
1 --2' 1rt

1

G«) -
-I'

D .. -

_1 (G«() d( 1\ d(. 21ri 1- z(

ziI::c.() =

io

z so that

F(z) = -zF(z),

(3.25)

z E ']['.

We now express the Dirichlet norm of an analytic function J E V(J[))) in terms of its boundary values 1*, which for convenience we denote simply by J With J(z) = L:'=o f(n)zn we have

1/(0)1' + /.11'(z)I' dz ~ 1[(0)1' + /.I~ n[(n)zn-,i' dz

/1(0)/2 + f /n1(n)/21Izn-112 dz ~ fen + 1) /1(n)/2. n=l

D

n=O

3.4. Other corona problema

49

Now we compute the boundary expression (note that r1L'::,.t corresponds to one derivative)

r 1 (1If(O+t)-f(O)IIL2(,][,»)2 dt t! t

Jo

r 1 r271"If(O+t)_f(O)12dOd; t

=

Jo Jo

r L I[(n) I le

Jo

1

2

ex)

int -

112

n=O

t;

d

2

00

~ Lnl[(n)1 . n=1

This suggests that for 0 ::::: a < 1 and any trigonometric polynomial define

{I/, II' + J.' (11/(9

11/11 BHT) ~

+I)

~/(9)

II£'(T)

)'~

r-

f

on 'll' we

and denote by B!J ('ll') the completion of trigonometric polynomials under this norm. Clearly B!J ('ll') is a Hilbert space with the inner product (f, g) u given by

(i f) (1 g) + 11 1271" (f(O + t) - f(O))(g(O + t) - g(O))dO tl:~U. Remark 3.9 The space B!J('ll') consists of functions on the circle 'll' having a derivatives in L2('ll'), while in our notation the space B!J(lIJJ) consists of analytic functions on the disk lIJJ having 1 - a derivatives in L2(lIJJ). The above calculations show that for

f

analytic,

f

E

1J(lIJJ).

It is easy to see that the operator of multiplication by z, which is the backward 1 shift operator on Fourier coefficients, is a bounded operator on Bi ('ll'). Thus by

(3.25) and the fact that

F(z) is analytic, it suffices to establish that

i I!F(Z)1

2

dxdy ::::: C

i

IG(z)12 dxdy.

For this we compute

I!F(z)1

=

12~i

l

(1

~(;~)2(d( A d(1 : : :

L

K(z,() IG«)I d.;dl1,

where K(z, () = 7I"ll!z<12. Using the estimate

1 1m ( II)

IwI2)_.! _ 2 2 dw ~ C(l _lzI2)-~ 11-wzl

we obtain that the "Schur function" h«) = (1 _ 1(1 2 )-! satisfies

L

K(z, ()h«)d( ::::: Ch(z).

A simple case of Schur's test now yields (3.26).

(3.26)

50

3. The Corona Problem

Theorem 3.10 (si'Tnple SChUT'S Test) Let (X, /1) be a measure space and H(x, y) be a nonnegative symmetric kernel. Define

=

Tf(x)

L

H(x, y)f(y)d/1(Y)

Then T is bounded on L 2(/1) if and only if there is a positive function h E L 2 (/1) such that Th(x)

L

=

H(x, y)h(y)d/1(Y)

:s;

/1- a.e.x E X

Ah(x),

Moreover, if A is the least such constant, then IITlloperatoT

= A.

Proof: We have

L

ITf(x)12 d/1(x)

<

<

L(L L(L L ~r:;

H(x, Y)h(Y)d/1(Y))

A

(L

H(x, Y)h(X)d/1(X))

< A2

hey)

d/1(Y)

Conversely, if T is bounded, choose

=~ 00

hex)

(

"f

= A2

H(x, y)

~r:)2 d/1(Y)

L

f(y)2d/1(Y)·

f > 0 in L 2 (/1), 1

IITII

)n

~r:; d/1(Y)) d/1(x)

"f

> I, and define x E X.

Tnf(x),

Then

and

~ ("f I~TII) n T n+ f(x) 1

Th(x)

"f

IITII ~("f ItTl1 )nTn f(x)

"f IITII (h(x) - f(x))

<

"f

IITII hex).

Altogether we have

IIF l'fll! :s; B2 ('f)

cliff l'fll

1

B22

('f) ~

Ilffll

W"

2(~)

:s;

cIIGII£2(~) :s; cIlfllv'

Remark 3.11 Although we will not need this, we point out that for general

f we have the "trace inequality" ,

3.4. Other corona. prabl......

51

To see this, we need to define the space ences. for 0 ~ a < 2 we can define

II/II B "

...)

~ {I.£ II' +

B2 ('Jr)

1.' CI

(8+ t) - 2 /

for a 2 1 using higher order differ-

(8;. + 1(8 -

t) II P(T) ) '

~

r' 1

(f,g}u.

and the corresponding inner product It can be shown that this expression for the norm coincides with the previous expression when 0 ~ a < 1. Then with ie ) = ie ) = fez), we compute using the second difference inner product and Cauchy-Schwarz that

fr(e

f(re

Ilf ITII:!eT)

=

! (fr,fr)~ dr =

{!I,!I}~ = fo1

2Re

fol (!fr,fr)! dr

folll!frt~(T) !1:rfrt~(T) dr 10t \1 !_frl!2B~(Y) dr + 10t IIfrll~l(T) dr

< 2 <

2

(fl

< C

10 rVf(z) 12 dxdy.

Thus we have one solution b1 by IIfliv

= F+h to (3.12) whose LOO('Jr) norm is controlled

+ IIflloo' and another solution b2

1

= F whose Bi ('f) norm is controlled by 1.

IIfliv + IIfll oo · What we want is a single solution b whose L oo and Bi norms are Simultaneously controlled by IIf\lv. For this we use Corollary 4.39 (p. 83) and Remark 441 below (see also Corollary 3.1 of Peller and Hruscev [34]) that says the best approximation operator A acts boundedly on the space exists a unique ho E BMOA('Jr) such that IIF + holloo

=

IlhoIlB!eT) ~

B~ ('Jr): i.e. there

inf{IIF + hlloo : hE BMOA} ~ C(ll f llv + IIfll oo )' CIIFIIBj(,U')

+ ho solves (3.12) with IIblloo ~ C(lIfliv + IIflloo) and IIbll! ~ IIFI!, + IIholl! ~ C IIFIl ~ ~ C(lI f llv, II f lloo)· B2 (Y) B2 (T) B2 (T) B2 (T)

Thus b = F

Thus the functions gj given by (3.10) in the previous subsection are analytic with

Bl

boundary values in LOO('Jr) n ('Jr), hence gj E BOO(D) n D(D). This completes Nicolau's proof that the Banach algebra BOO(D) n D(D) has no corona. 3.4.2 The algebra MV(D). We now sketch Xiao's proof in [52] of the corona theorem for the multiplier algebra MV(D) of the Dirichlet space D(D) (see Tolokonnikov [50] for the first proof of this). Recall from the previous subsubsection that 1 Bi ('f) is the Besov space normed by

IIhll

1

B;(T)

=

{ll

'll"

f

!2 +11(!lf(O+t)-f(O)!lL2(T»2dt}1 d 0

3. The Corona Problem

52

Moreover, arguing with Plancherel's Theorem as above we obtain,

l/hlJ,

B2 (T)

~ !/lP'hllwl 2(D) = ( JrD jVlP'h(Z)/2 dXdY) , , 1

if lP'h denotes the Poi&son extension of h to the disk Jl} Define the space Xi (1f) by requiring that not only is the measure d/lh(z) = IVlP'h(z) 12 dxdy finite, but also a V-Carleson measure:

IIhllXi(T)

= IId/lhlJv-Ca.le,on'

where for any measure /l we define

_, Up Ig!2 d/l)~

II/lllv-carieson = :~g

IIgliv

to be the norm of the embedding V c L 2 (1f). 1 1 Thus X 22 (1f) c Bi (1f) and h E Mv if and only if h E HOC has boundary values 1

1

1

in Xi (1f). However, we do not know at thib point if Xi (1f), like B} (11'), is an 1lspace or not, so that we cannot use Nicolau's argument here In&tead we will follow Xiao's proof that uses the continuous version of Jones' constructive interpolation, Theorem 2.11. Here is a sketch of the details First we use Gelfand theory as usual to reduce the proof to finding corona solutions for corona data. Then we use the Koszul complex in lD> to reduce matters as above to solving the d-bar equation (3.12) {}

azb(z)

= G(z),

Z

E Jl},

where d/l(z) = IG(z)1 2 dxdy is a V(lD»-Carleson measure In particular, from the previous subsection we see that F(z) = ~£<2 d( /\ as in (3.20) satisfie& F = G and the Dirichlet estimate

de

2!.i J JD

:z

/IF

jTI/BJ ('Ir) S =

C

11F1J~1l 2(1D)

C l'G(z)12 dxdy + C

In

/BG(z)j2 dxdy

S C IIfll~ .

To obtain the required multiplier estimate

IIF ho/l x } (T)+Loo + IIF IT/lL'X(T) S C IJ f l/!1v ' we use instead Janel:,' operator in Theorem 2.11 function S(v)(z) = where u

With dv(z)

= G(z)dxdy, the

JL

K(u, z, ()dv«)

= - JvL . and ~

K(u,z,() == 2i 'If

2

1- 1(1 exp (z - ()(l - (z)

{J Jr

1wl:2I<1

+~z +1 +~() dU(W)} '

(_1 1 - wz

1 - w(

satisfies :ZS(v) = Gin lD> and IIS(v)lIoo ::::; C !!vl!car S C !!/lllv-car where d/l(z) = !G(Z)!2 dxdy is a V-Carleson measure.

53

3.4. Other corona problema 1

We must now also show that S(v) dp,s(v)

==

Xi (T), i e. that IVS(v)(z)1 2 dxdy ITE

is also a 'D-Carleson measure. Of course /zS(v) = G in Jl} and we obtain immediately that! tz-S(v)(z)!2 dxdy = dp,(z) is a D-Carleson measure. Rather than attempting to use the singular Beurling transform B to estimate -IzS(v), we instead use the following clever device of Xiao introduced above (but used first by Xiao here) that greatly ameliorateb the singularity. We introduce the function S(v)(z) =

Jl

K(a,z,()dv«)

where K(a,z,()== 2i 7r

I-~122exp{J

11 - (zl

{

Jlwl~I<1

(_l+~z + l+~()da(W)}. 1 - wz

SinC'e K(a, z, ()

K(a,z,()

=

z - (" l-z("

1 - w(

«)

= 'P z

is the autormorphism of the disk that interchanges 0 and z, we see that S(v)(z) = zS(v)(z) for z E T (.pz«) = = z for z E T) Thus it suffices to show instead

z(t::?

that \VS(v)(Z)\2 dxdy is a D-Carleson measure. The advantage is that K(a,z,() has a much milder singularity than K(a, z, () Now the function S(v)(z) behaves essentially like the holomorphic projection r1v(z) where, more generally in higher dimensions,

r.g(z) == { g(w)(1_lwI2)S dw. . JBn (1 - w . z)l+S Lemma 5 16 in [5] shows that IVr1 (v)(z)1 2 dxdy is a D-Carleson measure since v is. An inspection of the proof shows that the argument extends to also prove that IVS(v)(z)1 2 dxdy is a V-Carleson measure Alternatively, one can use a theorem of Rochberg and Wu for thib purpose as is done in [52] This completes our sketch of Xiao's proof that the multiplier space MV(D) has no corona. 3.4.3 Further results. Recently, Arcozzi, Blasi and Pau have solved the corona problem for all of the multiplier spaces MB~CD) in the disk with 0 ~ a < ~ and 1 < p < 00 using Jones' solution to the d-bar equation ([3]) An interesting new method of attack on corona problems can be found in Treil and Wick [49] that is based on a projection lemma of Nikol'skiY. See also the references there for work on the matrix corona problem by Treil and others. In Subsection 5.5 below we will give a partial result of Ortega and Fabrega [33] toward the corona theorem for the multiplier algebras MB~(B,,) in higher dimensions n > 1 when 0 ~ a < ~, namely the case of two generators The actual corona theorem for the multiplier algebras MB~(Bn) when 0 < a ~ ~ and n > 1 has just recently been obtained by three of the participants in the lectures at the Fields Institute for which these notes were prepared ([18]). In particular this includes the corona theorem for the multiplier algebra of the Drury-Arveson Hardy space H2. 1 < n . .n When n > 1, the cases 2" < a - "2 remam completely open as well as all cases WIth

8. The Corona Problem

P i= 2. It is a longstanding open problem to determine whether or not Hoo(llJ n ), the multiplier algebra of the classical Hardy space H 2 (lffi n } = (lffin) , has a corona. See Varopoulos [51] for partial results, and Ortega and Fabrega (32) for related results.

Bl

CHAPTER 4

Toeplitz and Hankel Operators The sequence of functions B = {zn}~=o is an orthonormal basis for the Hardy space H2 on the disk. A linear operator B mapping into H2, whose domain of definition includes the basis B, is called a Toeplitz operator if its matrix representation [B] = [(Bzk, ZJ)]i,k<,:O relative to the basis B is constant on the main diagonals, i.e. there is a sequence of complex numbers {bn}~=_oo such that J, k ;:::: 0

(Bzk, zi) = bk-i,

The sequence of functions B_ = {Z"} ~=l is an orthonormal basis for the orthogonal complement H:' = (H2).L of H2 in L2(T). A linear operator A mapping into H:', whose domain of definition includes the basis B, is called a Hankel operator if its matrix representation [A] = [(Azk,zi)k:~:l,k~O relative to the bases B and B_ is constant on the cross diagonals, i e. there is a sequence of complex numbers {an}n~l such that (Azk,zi) = ak+J' j;:::: 1,k;:::: o. These definitions are motivated by the matrix representations of multiplier operators M", on L2(T) where cp E Loo(T}. Indeed, if cp(ei6 ) has Fourier series LnEZ;P( n )eine , then the matrix representation of M", relative to the basis {eint } nEZ of L2 (1I') is [M",]

=

[(M",zk, zi)]J,kEZXZ

=

[;P(J - k)]j,kEzxz.

Theorem 4.1 A Toeplitz operator B is bounded if and only if there is cp E Loo(T) with;P( -n) = bn for nEZ, in which case \lBIi = IIcp!loo· Theorem 4.2 Let A be a Hankel operator. The follOwing are equivalent (the equivalence of 1. and 2. is Nehari [26], while the equivalence with 3. is Fefferman's Theorem 4.4, 45). 1. A is bounded. 2. There is cp E Loo(T) with ;p( -n} = an for n ;:::: 1. 3. E~=l anzn E BMO(lI'). Moreover, in the event that A is a bounded Hankel operator, then there is cp as in 2. such that IIAII

=

IIcpllL'>o(T)

= distL"o(Cp, BOO) ~

Ilf

=

anznll

n=l

IJP-cpIlBMO(T)

(4.1)

BMO(T)

where P_ is orthogonal projection from L2 to H:'

See the next subsection for the definition of BMO and Fefi"erman's Duality Theorem 4.4 (p. 59). The bounded function cp in Theorems 4.1 and 4.2 is called the symbol of the Toeplitz operator B and Hankel operator A respectively. The symbol

56

4. Toeplltz and Hankel Operators

of a Toeplitz operator is uniquely determined (since all of its Fourier coefficients are prescribed), while that of a Hankel operator is only determined up to a bounded holomorphic function (since only its negative Fourier coefficients are prescribed). Note that with P+ the orthogonal projection from L2 to H2 and P_ the orthogonal projection from L2 to H'3..., we have

Indeed, for k

~

0 and j

(p_(cpzk),zj) = (

~

1, we have

L

ij3(n)zn+k,zj) = ij3(-k - j) = ak+j = (Azk,zj), n+k
=

A is Hankel if and only if AS =

BS* on H6;

P_SA;

where S is the unilateral shift on H2 and S is the bilateral shift on L2(11') given by (C.14) in Appendix C. One easily obtains these assertions by testing the commutation relations on the appropriate basis functionf>. Finally, note that a Toeplitz operator Tep = P+Mep is never compact unless cp = 0: if f E H2 is such that Tepf -=I- 0, then

IITep(znf)IIH2 = IJP+(zncpf)II H2 ~ IJP+(cpf)II H2 = IIT 0

IT

for all n ~ 1 while zn f -+ 0 weakly in H2 as n -+ 00: (zTt f, zR-) = zn-e f (z )dm( z) = 0 for n ~ + 1, ~ O. On the other hand, Hartman's Theorem, Corollary 4 11 (p. 67), shows that the Hankel operator Hep = P_M

e

cp E Hoo(1I')

e

+ C(1I')

Remark 4.3 An instructive way to see that the Toeplitz operator Tep is not compact when cp -=I- 0 is to notice that with ducing kernel for H2,

(T
£) =

(cp£,

£) = iTf cp 1£1 2 =

21 IT

£«() =

1-31 the normalized repro-

r cp(ei9 ) 11 -_~zI2 2 ze

L"

1

d() = 21 IPcp(z) ,

z9 1

IT

where IPcp is the Poisson kernel of cp Let z = re i9 . If L is the rank one operator Lf = (I, g) h where g, h E H2, then by Theorem C.IO (p 187),

(L£, £) = ( £, g) (h, £) = (1 - r2)g(rei9)h(rei9)

-+

0

as r -+ 1 for a.e. (J. By linearity this persists for finite rank operators L, and hence for compact operators L as well using Remark A.47. Thus if Tep is compact, then lilIlr-+llPcp(rei9) = 0 for a.e. 9, and the theory of the poisson integral shows that cp == O. In fact this argument shows that IIT",lIess ~ IIcplioo (see Definition 4.8 (p. 66», and from Theorem 4.1 we then conclude lIT",U = IITcpllesB = IIcplioo'

4. ToepUtz and

Hankel Operato....

57

Proof: (of Theorem 4.1) Let B be a bounded Toeplitz operator. For each nEZlet

Bnf

= zn BP+zn f,

Then Bn is bounded on L2 and

(BnZ k ,Zt) = (B Z n+k ,Zn+£) -- bk-£,

n+k

~

O.

Thus {Bn}~=l converges to a bounded operator C in the weak operator topology Indeed, if P(z) = L:lkl~N O!k zk and Q(z) = L:lll~N f3£zl are trigonometric polynomials on 'JI', we define C on trigonometric polynomials by

Then

II Gil =

sup II P IIL,2=IIQII L 2=1

I

\(CP, Q)P(T) :s; lim sup II B nll :s; IIBII, n-too

and so C extends uniquely to a bounded linear operator on L2 ('JI') satisfying

=

lim (Bzk+n, zHn) n-too

= bk-£,

for k, i E Z In particular,

Moreover we have Indeed, for i E Z we have

(Czn,zl)

= bn-I = (C1,z£-n) = (z nC1,zl).

Thus with r.p = C1 we have Czn = r.pzn for all nEZ, hence Cf = r.pf for all f E L2. Since the pointwise multipliers of L2 are isometrically isomorphic to Loo, we conclude that IICII = 1Ir.plloo. Altogether for I E H2 we have

BI = BP+I

= P+CP+I = P+r.pl,

and

= IJP+r.pllb :s; 1Ir.p1100 II/lb = IICllllflb IIBII :s; IICII, and hence that IIBII = IICII = 1Ir.p1l00· IIBflb

This shows that

Proof: (of Theorem 4.2) By Corollary C.8 in Appendix C the unit ball in HI equals

{f = gh: g,h E H2 with IIgIi H2, IIhllH2 :s; I}. Thus the products gh form a dense subset of the unit ball in HI as 9 and h roam over all polynomials in z in the unit ball of H2. Now (Azk, zj) = ak+j = (AI, zk+ j ), and

4. Toeplitz and Hankel Operato....

58

so (Ag, zh) we have

=

(AI, gzh) for all polynomials 9 and h. Thus with /, g, h polynomials,

=

HAll

sup

I(Ag, zh)1

sup

\

IIg1l H2,lI h 1lH29 IIg1lH2,lIh1lH29

f

iT

(42)

Z9h(AI)\

IiTf Z/(Al)1 sup IIrf f(AI)I·

sup

II/11H19

IIfIl H 19,/(0)=0

Suppose the Hankel operator A is bounded. Then by the Hahn-Banach Theorem the bounded linear functional f ~ f(AI) on HJ extends to a bounded linear functional on £1 with the same norm IIAII. Thus there is

IT

h

f(AI)

Note also that IT/(AI)

=

h

f
= I]:fP-

=

E

HJ, and we see that

11P-
by Fefferman's Duality Theorem (see Remark 4.5 below; remember that polynomial in (4.2». Now we have

ip(-n) =

f

is a

h

zn

for n ~ 1. From this we see that P_

IT

HAil

= 1I<>(
OO

)

= IJP-
Ilf anznll n=1

~

11P-
(HJ)O

Ilfan:znll BMO . n=1

Using Al = P_

We say that a function b E £2(11') on the circle has bounded mean square oscillation, or simply bounded mean oscillation, if

IIbli.

~ fc!: { I~I 1, ib(e"') - I~I 1, { : }I <

00.

4.1.

HI - BMO

duality

59

Such functions are exponentially integrable, hence close to being bounded (Corollary B.ll in Appendix B). The space BMO = BMO(1r) of functions b having bounded mean oscillation is a Banach space when normed by 1fT + I!bll •. We also define the space of analytic BMO functions by

II

BMOA(1r)

= {b E BMO(1r) : ben) = 0 for n < O}

Typical examples of BMO functions are bounded functions and log 101. The following famous theorem of Charles Fefferman (Theorem 1 of Chapter IV in [46]) identifies the dual of HJ(1r) as BMOAo(1r), the subspace of BMOA(1r) consisting of functions that vanish at the origin.

Theorem 4.4 (C. FeJJerman) HJ(1r)* = BMOAo(1r) under the pairing (h, b) H2 for h a trigonometric polynomial. Moreover, if b E H8(l1) and ~ is the linear functional Abh = (h, b) H2 on polynomials, we have

II A bIlHJ(T)' ~ IIbll* ~ lIJ-LbIlH2-carleson'

(4.3)

where dJ-Lb(Z) = Ib'(z)1 2 (1 -lzI 2 )dxdy.

Proof: It suffices to prove (4.3), which we accomplish by proving the three inequalities II A bII HJ (T)'

< C IIJ-Lb II HLCarleson , IIbIIBMOAo(T) < C IIAbIlHJ(T)' , IIJ-Lb II H2-Carleson < C IIbIlBMOAo(T)

(4.4)

To prove the first inequality in (4.4), we use the following consequence of Corollary C.8 (p. 187): as polynomials g(z}, h(z} with g(O)h(O) = 0 roam over the unit ball of H2, the products g(z)h(z) are dense in the unit ball of HJ. Recall the equivalence of norms in H2 (lI}) and

Ib(O)12 +

1

Bi (lI}):

h

lb(e it )

-

b(O) 12 dt

~ Ib(O)1 2 +

L

Ib'(z)1 2 (1-lzI 2 )dz.

(4.5)

We need the following exact version of this equivalence, (4.6) which follows from Green's Theorem

J

l(V8.U-U8.V)dXdY =

fan (v: -U::)dS,

fzr

using u(z) = lb(z) - b(O)1 2 , v(z) = log and n = B(O, r) \B(O, c), and then letting r -+ 1 and c -+ O. Polarization of (4.6) and 1(0) = 0 yield

~f = (f,b}H2

=

1 21r

f

it--

Jr/(e

)b(e·t)dt

1 =:;1 JfD l(z)b'(z)logJ;Tdz.

60

4. ToepUtz and Hankel Operators

Then using the definition dji:/,(z) polynomials, IAb!l

=

=

Ib'(z)1 2 Iog rhdz, we have with!

gh,g, h

110 (g'(z)h(z) + g(z)h'(z»)b'(z) log 1~ldzl .i

<

=

(1 Ig'(z) 12 l~ldZ) (1 (L Ig(z)(z)1 2dji:/,(Z») (1 10g

2

...

Ih(z)(z)1 2 d/it,(z»)

.i 2

IIgll B1 (I[)

2

Ih'(z)1 10g

211llbllH2-Carleson

~

211llbllH2-Carleson llgIlH2(D) IIhIlH2(1[)

since 1 - lzl2 ~ log

... 2

B! (D)

<

IIhll

I~I dZ)

2

= 2 IIllb II H2-Carleson II/I1HI(D)

rh for ~ :::; Izl < 1 easily yields IIllbllH2-carleson

~

IIllbllH2-carleson

To prove the second inequality in (44), we use the Hahn-Banach Theorem A 36 (p. 162) to extend Ab to a continuous linear functional (with the same norm) on the real variable analogue Re HI of the Hardy space HI. Here Re HI denotes the complex linear span of the real parts a of HI functions on the circle normed by

lI a liReHI

=

~ (lI a ll£l(T) + llall£l(T)) ,

where

a=

-i(P+a - P_a - a(O»

is the conjugate harmonic function of the harmonic function aj then a + ia is holomorphic and a(O) = O. In particular, note that II/I1HI = Il/l1ReHl for I E HJ. o Denote the norm-preserving extension of Ab from HJ to Re HI by
J


iJ

kfE,

IE L2

We now claim that (4.7) To prove (4.7), we say that a function a E L2(T) is a ReHI-atom adapted to an arc 1 if

supp a eland

j

a(O)dO

= 0 and

j

1a(O)1 2 dO:::; 111- 1 .

Let CI be the center of I. Note that if a has Fourier series LnEZ a(n)ein6 on T, then the conjugate function a has Fourier series Ln;>"O -ifnya(n)ein6 • A routine calculation shows that is given away from the interval 1 by the convolution integral

a

a(O)

-t) a(t) 21rdt JI(cot (()-2-

1 (0; {cot

t) - cot (() ~

CI)} a(t):;,

()ET\!.

4.1. H1 - BMO duality

61

= rei6 ,

Indeed, we simply note that for z eit + z e it _ z =

.2rsin(9 - t) ze it l2

1- r2

11 _ zeit 12 + z 11 -

has real part the Poisson kernel, and with r = 1 the imaginary part above is 2sin(0 - t)

= 2sin(¥)

1-2cos(0-t)+1 Thus we have

i

la(O)1

~: =

cos(~)

= cot

2sin2 (02 t )

(0 -2 t) .

III~ (/la(0)1 2 dO)!

/la(O)I: ::;

::; 1,

and

f la(O)1 dO

In:

27r

::;

+

f la( 0) I dO

i21

27r

f

i'f\21

la( 0) I dO 27r .1

< 12II! ( f la(O)12 dO)

i2I

+ f

2

27r

flcot(O-t)-cot(O-CI)lla(t)ldtdO

i'f\21 if

27r 27r

1

<

12II~ (i la(O)1 2 : : ) 2 +f1a(t)l{f

il

i'f\21

< v2 + C

Clt-CfldO}dt 10 - cII2 27r 27r

f la(t)1 dt ::; C if 27r

It follows that a E Re HI with

lI a ll ReH

=

l

~(lIaIlHl + BalIHl) ::; C.

Recall that

Now let f roam over all Re HI-atoms a adapted to I. Using that both a and h - hI have mean zero on I, we compute that

a

:5:

ada~~~

to f

sup

a adapted to I

1/

a(h - hI)I

= a

ada~~~

IIhll* :5: C 1Iq> II (Re Hl)*

= C IIAbll(HJ)* .

Similarly the linear functional ~ defined by

hjh i lh, = -

III

Iq>al ::; 1Iq>lIl1allReHl ::; C 1Iq>1I.

Taking the supremum in 1 shows that

~ f == q>1 =

to

f

E £2,

alii

62

4. Toeplitz and Hankel Operators

is bounded on ReHl with norm that

1I~ll

=

1I<.p1I, and the above argument now shows

Ilhll* ::; C 11~II(ReHl)* = C II A b II (HJ)* .

Altogether we have that IJP+hll* ::; of hand

h.

c II AbII (HJ)*

since P+h is a linear combination

Thus P+h E BMOA and for f E HJ a polynomial we have

lfP+h=

lr 1li

= <.pf = Abf = llb

It follows that b = P + h - IT h E B M 0 Ao and that the second inequality in (4 4)

holds. Finally we turn to proving the third inequality in (4.4) following equivalence for b E H2 (lI);

IIbll~ ~ sup f

wEnJT

Ib(e it )

-

We begin with the

b(w)1 2 Pw(t)dt,

(4.8)

where

eit + z 1 - IwI 2 2 Pw(t) = Re -.-- = e't-z 11-w(1 is the Poisson kernel This is a straightforward consequence of standard estimates on the size of the Poisson kernel (for details see Theorem 1.2 in chapter VI of [20]). Next, we recall that the equivalence of norms (4.5) yields

i

Ib(e it )

-

b(0)1 2 dt

~ L1b'(z)12 (1-lzI 2)dz.

Note the following identities for IPw«() = 1~1c;:

l-lwl 2

IP~«()

(1 - W()2

( and IP~(IPw«(»)IP~«() = 1,

(1 -

1(1 2 )(1 -

Iw1 2 ) 11 - (w1 2

Now replace b by bolPw in (4.5), and then make the change ofvariables eit = IPw(e i8 ) and z = IPw«() respectively in the left and right sides of (4.5) to obtain both

lr

Ib 0 IPw( eit ) - b 0 IPw(O) 12 dt

and

i lr

Ib( ei8 )

-

b(w) 12IIP~( eiO ) 1dO

Ib(e iO )

-

b(w) 12 Pw(O)dO,

1

1(b 0 IPw)'(z) 12 (1 - Izl2)dz

llb'«()12IIP~(IPw«(»12 (1-IIPw«()12) IIP~«()12 dz =

Ib'«()1 2 (1 - 1(1 2 )(1 ~ Iw12) d( In 11-(zl

f

=

f

in

Pw«()dt-tb«()'

4.1. HI -

BMO

duality

63

Thus we have

IIbll~ ~ sup

( Pw«()dJ.tb«()·

WEIDJID

Once again standard estimates on the size of the Poisson kernel (for details see Lemma 3.3 in chapter VI of [20]), together with Carleson's Theorem in Remark 2.7 (p. 18), show that

This establishes the third inequality in (4.4), actually with approximate equality.

Remark 4.5 From the above argument we also have

(ReH1('Jl'»* = ReBMOA('Jl') with equivalence of norms, where ReBMOA('Jl') is the complex linear span of real parts of BMOA('Jl') functions normed by

II f\ +

llbIIReBMOA(T)

= II bIIBMo(T) + IlbllBMO(T) .

Since the conjugate function is bounded on B M 0(11'), we obtain that Re B M 0 A('Jl') = BMO('Jl'). To see that the conjugate function is bounded on BMO, we extend the equivalence in (4.8) to hold for b E L2('Jl') with b(w) = JP>b(w) given by the Poisson integral. Then it is enough to prove that

But this reduces to the case w = 0 after changing variables with the automorphism that interchanges wand 0, and this case is just IT

b E L2 with IT b =

o.

Ib(eit ) 12 dt

= IT

Ib(eit ) 12 dt

for

See Corollary 2.5 in Chapter VI of [20} for details.

Remark 4.6 Elementary functional analysis yields the characterization (ReH1(11'»* ~ LOO('Jl')

+ Loo('Jl').

Indeed, the embedding I: ReH1('Jl') -+ £1('Jl') EB L1('Jl') defined by If = (/,1) is continuous with closed range, and so Proposition AAO in Appendix A shows that

(L1('Jl') EB L1(11'»* j(IReHI ('Jl'».L (L 00 ('Jl') EB L 00 ('Jl') ) j {(h, h) : h E L 00 } LOO('Jl') + Loo('Jl'), since

IT lh = - IT Iii implies (IRe HI ('Jl'».L = ((g,h) E LOO('Jl')

+ LOO('Jl') :

1

= {(g, h) E LOO(T) + £ClO(11') : 9 -

Ug + lh) =O,f E ReHI('Jl')}

h = OJ.

64

4. Toeplitz and Hankel Operators

4.2 Compact Hankel operators

We begin with Kronecker's characterization of finite rank Hankel operators. Theorem 4.7 Let Hip be a Hankel operator with symbol cp. Then the following conditions are equivalent· 1. Hip is of finite rank, 2. ker Hip = b1 H2 for a finite Blaschke product b1 , 3. ~cp E HOO for a finite Blaschke product ~, 4. P _ cp is a rational function. Moreover, we have the following equalities: rank(Hip) = #{zeroes of b1 } = #{poles of P_cp}. Proof: First we note that for f E H2 and g E H'3..., (P-cpf,g)

(Hipf,g)

=

(cpf,g)

=

(f,<j5g)

=

(f,P+<j5g)

(I, zP_zcpg) = (I, zHipzg) = (zHipzg,/) Thus f E ker Hip if and only if 1 1. range(T) where Tg = zHipzg. Now zg ranges over H2 as 9 ranges over H'3..., and since multiplication by z is an isometry, we see that rank(Hip) = dim range(Hip) = dim range(T) = dim(ker Hip)-L. Now ker Hip is invariant under multiplication by z (see (4.27) below) and so by Beurling's Theorem C.16 (p. 193) in Appendix C, ker Hip = bH 2 for an inner function b. Thus rank(Hip) = dim(bH2)-L.

We now claim that dim(bH 2 )-L is finite if and only if b is a finite Blaschke product Indeed, if b(z)

=

N

N

n (l~;Z)mn = n bAn (z)m,. is a finite Blaschke product, n=l n=l

n

N_ bm" H2 where then bH 2 = n n-l An

(b~n H2)-L

{J E H2 : (b'!:"ng, f) =

v'!1n-l{bt k } = J=O

An An

= O,g E H2}

Vm,,-l { J=O

(410)

(An - z)j } (1 _ AnZ)J+l '

and kA,,(z) = l-~nz is the reproducing kernel for H2 at An ElI}. This is obvious if An = 0, and the general case follows by composing with the automorphism CPA..{z)

= bAn (z) since bAn 0

CPA..{z)

=

N

n b~n we have

z. Thus for b =

n=l

(4.11) and if ker Hip

= bH 2, then N

rank(Hip) = dim(bH 2 )-L =

L

mn,

n=l

the number of zeroes of b counting multiplicity in lI}. Conversely, if dim(bH 2 )-L is finite, then (bH2)-L is spanned by the rootvectors of S*, since by Beurling's Theorem C.16, the orthogonal complements (bH2)-L are the S· -invariant subspaces of H2. Recall that if S* on (bH2)-L is represented by

4.2. COlnpact Hankel operators

65

a complex m x m matrix A in Jordan canonical form, the rootspaces of A are the subspaces ker(A - AnI)mn, 1 :::; n :::; N, where An are the eigenvalues of A and mn is the size of the largest Jordan block with diagonal element An. The elements of a rootspace are called rootvectors. However, S*J =).J with J E H2 if and only if [en) = Ai.(z) = l!Xz is the unique (up to a constant multiple) corresponding eigenvector. More generally,

°

ker(S* - "XI)m = V,?="(/ {

zi.} ,

(4.12)

(1 - AZ)J+l

follows using the identities ker( S* - "XI) m

H2

(S - AI)mH2

e (S -

.\I)mH2,

{J E H2 . J(j)(A)

= 0,0:::; j :::; m

-I},

(d~Y (f, k>.) (J, (d;Y 1 ~ ).z)

J(j) (A)

=

l' (/, (1 _

~z)i+l)'

O:::;j:::;m-l.

It follows that the Jordan canonical form of the restriction of S* to the invariant subspace (bH2)i. has, for each 1 :::; n :::; N, a single mn x mn Jordan block with diagonal elements An, and has rootspace V,?='O-l{ (l-::Z)Hl} corresponding to the eigenvalue An. Thus dim(bH2)i. = L:~=l mn and N k (S* (bH 2)i. -- Vn=l er -

~I)m" An

- VN

n=l vmnJ=o

-

1 {

zi

(1 _ AnZ)J+l

}.

(4.13)

However, we have equality of the linear spans in (4.10) and (4.12) when mn = m and An = A· m-l {

Vi=o

(A - Z)j}

(1 _ AZ)i+l

m-l {

= Vj=o

zj

(1- Az)i+l

Indeed, this is easily established by induction on musing

}

IAI < 1

Combining (4.11)

N

and (4.13), it follows that b =

IT

n=l

b';n is a finite Blaschke product. T~

Thus rank(H'f» < 00 if and only if there is a finite Blaschke product b such that ker Hcp = bH2 , i e. H'f>b = 0, which is in turn equivalent to cpb E Hoo or cp

= bJ

tfH

for some / E Hoo. Now if b =

= Q(z)

+ h(z) where hE Hoo

N

IT b';",... n=l

is a finite Blaschke product, and if

and (4.14)

is the principal part of the meromorphic function {" then

66

4. Toeplitz and Hankel Operators

since

2:':=1

I-tz

Z-\n

E H:'('Jr). Thus P-'P = Q is a rational function with mn = rank(Hcp) poles in the disk. Conversely, if P-'P is a rational func-

tion with principal part Q as in (4.14), then P -'P

=

Q and so 'P

N

Il

br;!:nn E H OO .

n=I

Definition 4.8 Given T . HI ---+ Hz a bounded linear operator between Hilbert spaces HI and H 2, we define the essential norm of T by

IITIless

== inf{IIT - KII : K is compact from HI to Hz}.

Sarason's Theorem will allow us to compute the essential norm of a Hankel operator. Theorem 4.9 The set HOO('Jr)

+ G('Jr)

is a closed subalgebra

01 the

Banach

algebra LOOer). Proof: The natural embedding j . G('Jr)jA('Jr) ---+ LOO('Jr)jHOO('Jr) is an isometry since LOO('Jr)jH=(T) can be identified with the second dual of G(T)jA(T). Indeed, (G('Jr)jA('Jr» * = HJ and (HJ)* = Loo(T)jH=(T). Thus the image j (G ('Jr) j A('Jr») of G ('Jr) j A('Jr) in the quotient space L = ('Jr) j H= ('Jr) is closed. If 7r : Loo('Jr) ---+ Loo('Jr)jHOO('Jr) is the quotient projection, then Hoo(T) + G('Jr) = 7r-I{)(G('Jr)jA('Jr))} is closed as well. Finally, the equality

H=('Jr)

+ G('Jr) =

closureLOO(T)(U~=oznH=('Jr)

shows that Hoo('Jr) + G('Jr) is an algebra. Here is a direct proof that the map J above is an isometry. We use the Poisson kernel f'/(re iO ) = Pr * 1(0). Since Ir ---+ I in L= for IE G(T) and Il/rlloo S 11/11= for I E H= ('Jr) we conclude that

distu,,"(j, Hoo('Jr)) = distLoo(j,A(T»,

IE G('Jr)

(4.15)

Indeed, S is trivial and for 9 E H= ('Jr) we have III - gll=

> :~ 1I(j - g)rll= > limsup{111 - grll= - III - Irlloo} r--->l

lim sup III - grll= ~ distu",(j,A('Jr). r--+l

From (4.15) we see that the natural embedding j : G('Jr)jA('Jr) is an isometry

---+

LOO('Jr)jHoo(T)

Here is the computation of the essential norm of a Hankel operator. Theorem 4.10 Let'P E L=(T). Then

IIHcplless

=

dist£,X'(T)('P, H=('Jr)

+ G('Jr».

Proof: If p is a trigonometric polynomial, then Hp is of finite rank by Kronecker's characterization, hence compact. Since the trigonometric polynomials are dense in G (T), (4.1) shows that if lEG ('Jr), then H f is a norm limit of the compact operators HPn where Pn ---+ I in G('Jr), and thus H f is compact. We then have from Nehari's Theorem,

dist Loo(1r)('P, H=('Jr)

+ G('Jr))

inf dist LOO (T) ('P -

fEC(T)

inf

fEC(T)

I, HOC> ('Jr»

IIH

67

4.2. CODlpact Hankel operato....

Conversely, let S be the forward shift operator on H2, i.e. SI(z) = zl(z). If K : H2 ~ H~ is compact, then IIKS"'II ~ 0 as n ----+ 00. Indeed, if K is onedimensional, then Kx = (x,j)g for some 1 E H2, 9 E H~, and so IIKS"'II = II(S*)"'/lIlIgll ----+ O. This persists for K with finite-dimensional range, and hence for compact K by Remark A.47 (p. 168) in Appendix A: if 11K - LII < c where L is finite-dimensional, then

IIKsnl1 ~

II(K -

L)snn

+ IILS"'II

~

c + o(n).

Thus we obtain

IIH", -

KII

~

II(H", - K)snll ;:::: IIH",snll-IIKsnn distLOO(T) (
> distv><>('lr)(
As a corollary we obtain Hartman's characterization of compact Hankel operators. Corollary 4.11 Let

v denotes the conjugate function of v given by v = -i(P+v - P_v - v(O».

Note that v + iv is analytic in]]) since v = P+v + P_v. Proof (of Corollary 4.11): The equivalence of assertions 1 and 2 is immediate from the above theorem. Indeed, if H", is compact, then

o = IIH", Iless = distL=(T) (
0= distvx>(T) (
+ G('JI'» =

+ G(T).

Conversely, if

IIH",lless ,

implies that H", is a norm limit of compact operators, hence compact. The equivalence of these assertions with 3 follows from (4.1) in Nehari's Theorem. Indeed, if 2 holds, say

1 = p_v = 2(v -

iv - v(O» E G(T)

+ c(1r).

On the other hand, by the boundedness of the conjugate function on BMO (see Remark 4.5 above), it follows that P_ is also bounded on BMO. Thus if 3 holds, say P_

IIH

=

~ ~

IIP-

o ----+ 0

68

4.. Toeplitz and Hankel Operato....

as r ~ 1. Hence compact.

H
is the norm limit of the compact operators Hpr'P' hence itself

Remark 4.12 Sarason has shown that

G('Jl')

+ C(j) == {u + 15· u, v E Gpr)} =

VMO('ll'),

(4.16)

where V MO('ll') is the closed subspace of BMO('Jl') consisting of those functions b on 'Jl' having vanishing mean square oscillation

IC~~~9 {I~I [lb(e

1

iO

) -

I~I [ br ~:} ~ ~ 0 as 8 ~ o.

Here is a brief sketch of the proof. Recall that we can extend the equivalence in (4.8) to hold for b E L2('Jl') with b(w) = lPb(w) given by the Poisson integral:

r Ib(e

IIbll~ ~ sup

it )

wEllJiIr

-lPb(w)1 2 Pw(t)dt.

Moreover, we can also show that b E V MO if and only if lim {sup ( Ib(e it ) -lPb(w)1 2 Pw(t)dt} r ..... l r::;lwl
iT

= O.

~

Thus G('Jl') c VMO ('Jl'), and from (4.9) we also obtain G('Jl') C VMO('Jl'). Conversely, we use the following characterizations of b E V M 0 in terms of continuity in the BMO metric:

bEVMO

if and only if 0lim IIb(O + t) - b(t) ..... 0 if and only if lim lib - Pr r ..... l

* bll* =

"* = 0,

(4.17)

O.

Now if bE VMO, Remarks 4.5 and 4.6 above show that b = a+ul +U2,

where a is a constant and Ul, U2 E LOO('Jl') with lIujlloc S G IIb/i" (this inequality is why the constant a appears). By (4.17), there is r < 1 such that

lib - Pr

* bll.

1

< 2"" bll*

Now let vY> be the corresponding Poisson integral of Uj:

vF>(t) Then vY> E G('11') and

Thus the remainder

satisfies

= Pr * Ul(t) and v~l)(t) = Pr * U2(t).

4.3. Best approximation

69

Repeating this argument with b

with vJ

E

=a +

R(I}

in place of b and then iterating we obtain

------L v~k} + L v~k) = a + 00

00

k=1

k=l

VI

+ 112

C('Jr) and

4.3 Best approximation

Here we follow Peller and Khrushchev [34]. We begin with Peller's characterization of Hankel operators in Schatten-von Neumann classes. We say that T . H 1 ~ H 2 is in the Schatten class gp if (4.18)

where sn(T) = inf{IIT - KII . K E Kn}

(4.19)

is the nth singular number (or s-number) of the operator T. Here Kn is the set of operators K . HI ~ H2 with rank not exceeding n Peller's Theorem is that a Hankel operator H

in the Besov space B; ('Jr) But before proving this, we take a small excursion into spectral theory for compact operators in order to identify the singular numbers as eigenvalues of a related operator, the positive square root of T*T. We note that if T is compact, then T*T is compact, self-adjoint and positive. If {Tn}~=o are the eigenvalues (necessarily nonnegative) of T*T listed according to multiplicity and in nonincreasing order, we claim that (4.20)

Note that Theorem A.51 (p. 170) in Appendix A tells us that the eigenvalues of a compact operator (even from one Banach space to another) are at most countable with 0 as the only possible limit point, and moreover that eigenspaces corresponding to nonzero eigenvalues are finite dimensional. However, this theorem is not very useful in reconstructing a compact operator from its spectral data, eigenvalues and eigenfunctions, as evidenced by the following example of a compact operator T on £2(z+) that has no eigenvalues and eigenfunctions whatsoever. Example 4.13 Let T = MS where S is the forward shift operator on Z+ and M is pointwise multiplication by {n~1 }~=o It is easy to compute that

rn{Xn}~=o =

{

Xn-m

(n + 1) ... (n - m

+ 2)

}OO,

n=O with the understanding that the terms corresponding to 0 :s; n < m are zero. We have IImll = (m~l)t. and so the spectral radius formula ([38]) yields sup

'xEa(T)

IAI =

lim IImll';' = lim ( m-+oo

m-+oo

1 )~ = 0 (m + I)!

4. 'Ibeplltz and Hankel Operators

Thus u(T) = {OJ, and since T is obviously one-to-one, we conclude that T has no eigenvalues at aIll Nevertheless, the operator To.T in the above example is pointwise multiplication by {(nt2)2}~=0, an operator that is much better related to its spectral data. Indeed, the eigenvalues are (nt2)2 with corresponding eigenvectors en, and these eigenvectors form an orthonormal basis for the Hilbert space £2(Z+), so that we can write 00 ( 1

To.Tx =

)2 (x, en) en, L -n+2

11.=0

Even better, we can reconstruct T from this data via the formula T = SVTo.T: 00

Tx =

L

1

--2 (x, en) Sen, 11.=0 n+

That such a reconstruction is possible for general compact operators on separable Hilbert spaces, is due to Schmidt, and we now turn to establishing this in the next subsubsection.

4.3.1 Spectral theory for compact operators and the Schmidt decomposition. The Banach algebra of bounded linear operators on a Hilbert space H is denoted B(H). An operator T E B(H) is normal if it commutes with its adjoint, i.e. T*T = TTo.. We will need the following simple observations regarding normal operators. Lemma 4.14 An operator T E B(H) is normal if and only if //Txll = IIT*xll for all x E H. Moreover, if T is normal, then 1. kerT = kerT*. 2. IfTx = ax for some a E C, x E H, then To.x = ax 3. If a and f3 are distinct eigenvalues ofT, then corresponding eigenvectors are orthogonal. Proof: If T is normal, then

= /IT*xIl 2 . (4.21) To.T-TT* satisfies (Sx,x) = 0 for all x E H,

IITxll2 = (Tx, Tx) = (T''Tx, x) = (TTo.x, x) = (To.x, To.x) Conversely, if (4.21) holds, then S = and so the two equalities 0=

o imply (Sx,y)

=

(S(x+y),x+y) = (Sx,y) + (Sy, x} , (S(x + iy), x + iy) = -i (Sx, y) + i (Sy, x) ,

= 0 for all x,y E H,

hence S = O.

Let {xn}~l be a collection of eigenvectors (possibly finite) of an operator

T E B(H), and let M be the closed linear span of {xn}~=l' Then TM c M, and by duality, T*(MJ..) C MJ.. «xn,To.y) = (Txn,y) = an (xn,y) = 0 ify E MJ..). However, if T is normal, then Lemma 4.14 implies that the vectors {xn}~l are simultaneously eigenvectors of T*, hence TO. M c M and T( M J..) C M J.. as well. In particular this applies to self-adjoint operators including for example To.T. This property will be crucial below for obtaining that the eigenvectors of T*T span H when T is a compact operator.

71

4.3. Best approximation

We will also need the following elementary properties of an orthonormal basis {Ua}"'EA in a Hilbert space H, where we say that {Ua}aEA is an orthonormal basis if (U a , up} = 8~ and the linear span of {U",}aEA is dense in H.

Lemma 4.15 Let {U",}aEA be an orthonormal set in a Hilbert space H. Then the following conditions are equivalent: 1. {u a } aEA is an orthonormal basis in H. 2. {Ua}aEA is a maximal orthonormal set in H. 3. L:aEA l(x,ua )l2 = IIxll2 for x E H. 4. L:aEA (x, u a ) (y, u a ) = (x, y) for x, y E H. It is an easy consequence of the Hausdorff maximality principle that maximal orthonormal sets exist (Theorem A.16 (p. 152) in Appendix A), hence every Hilbert space H has an orthonormal basis. If {Ua}aEA is an orthonormal basis in H, property 4 shows that each element x E H has a unique representation x =

L

(x,ua}ua . aEA The following theorem is a special case of the spectral theorem for normal operators. For comparison, we discuss the general spectral theorem (without all the proofs) in Appendix D below.

Theorem 4.16 Let H be a separable Hilbert space and suppose T E 8(H) is compact. Then T*T has at most countably many eigenvalues {An}~=l with at most one limit point, namely 0, and there is an orthonormal basis of H consisting of eigenvectors of T*T.

Proof: The operator T*T is compact and positive. Theorem A.51 (p. 170) in Appendix A thus shows that the set {An}~=l of eigenvalues of T*T is at most countable with at most one limit point, namely 0, and that An ~ 0 for all n ~ O. Each eigenspace En = ker(T*T - AnI) is closed and hence a Hilbert space. Thus En has an orthonormal basis {U;:!}aEA n • Now let M be the closed linear span of {U;:!}nEN,aEA". Then M is invariant for T*T, and since T*T is self-adjoint, the discussion following Lemma 4.14 shows that M.L is also invariant for T*T. Now we use the compactness of T in a decisive way to prove that M = H. Suppose, in order to derive a contradiction, that M.L i=- {OJ. Since M.L cannot be a subspace of kerT*T,

A=

sup

(T*Tx,x)

> o.

XEM-L,lIxlI:O:;l

Let Xk E M.L, IIxkll ~ 1 be such that lim (TXk' TXk)

k-+oo

= k-+oo lim (T*Txk, Xk) = A.

By compactness of T we may choose a subsequence, which we continue to denote by {Xk}~l such that TXk --+ Yo in norm, and by the Banach-Alaoglu Theorem for a separable Banach space (see Corollary A.45 (p. 167) in Appendix A), we may also assume that Xk --+ Xo weak star. But then TXk --+ Txo weak star implies Yo = Txo, and we have

I(Txo, Txo} - (TXk' TXk} I < I(Txo, Txo - Tx",} I + I(Txo = (II Txoil + IITxklD IIYo - TXkll

TXk, TXk} I

4. Toepiitz and Hankel Operators

72

tends to zero as k

-+ 00.

Thus Xo

M.l. with lIxol/

E

(T*Txo,xo) We claim that T*Txo

E

=

1 and

= A = I(T*Txo,xo)l·


Xo is an eigenvector ofT*T, and so Xo E MnM.l.

let h E M.l. with (xo, h) = 0 T*Txo = AXo + h, xQ+eh· 't t . M.l. and Then Xe =- l+e 2 lJhlJ2 IS a um vec or III A

> (Txe, TXe) = (

1 1 +c 2 IIhil

2) 2(Txo + cTh, Txo + cTh)

( 1 + 1IIhll 2) 2{A + 2cRe (Txo, Th) + c 2IIThll2} , C;2

implies that Re (Txo, Th) = O. If we consider Xc == 1:~tlt~~2 in place of Xe, this argument shows that 1m (Txo, Th) = 0 Thus (T*Txo, h) = 0 and we obtain

0= (T*Txo, h) = (AXo

+ h, h) =

A (xo, h) + IIhll 2 = IIhll 2 .

Thus we have proved that H = M = Span{U~}nEN,aEA" By assertion 3 of Lemma 4.14 we have u~ .1 u~ if m -I- n, and hence {U~}nEN,aEAn is an orthonormal basis of H consisting of eigenvalues of T*T. Now we return to proving (4.20), Le. that the singular numbers sn(T) of a compact operator T : HI ...... H2 are given by the eigenvalues of JT*T taken in nonincreasing order. Let A = .jT*T be the positive square root of T*T defined by 00

Ax=

L L

Fn(x,xn)x~,

(4.22)

n=OaEA ..

where {X~}nEN,aEAn is an orthonormal basis of HI and x::! is an eigenvector of T*T with eigenvalue Tn. For convenience let us write Un = Fn and {xn}~o for the orthonormal basis {X~}nEN,aEA" where T*Txn = T nXn. We need the following simple polar decomposition for a bounded operator from one Hilbert space to another, but only when T is compact, in which case the square root .jT*T can be defined by (4.22) above. Proposition 4.17 If T is any bounded operator from a Hilbert space HI to a Hilbert space H 2 , and if A is the positive square root of T*T {Theorem D.5 {p. 197} in Appendix D), then T=UA, where U is a partial isometry from HI to H 2, i e. an isometry from rangeA to rangeT and vanishing on (rangeA).l.. Proof: We have IIAxll2 = (Ax, Ax) = (A 2x,x)

= (T*Tx,x) = (Tx,Tx) = IITxll z

for all x E H. Thus the formula U Ax = Tx defines a linear isometry U from rangeA onto rangeT, which has a continuous extension U to a linear isometry from rangeA onto rangeT. Now extend U to B(Hl, H 2) by defining Uy = 0 for y E (rangeA).l..

73

4.8. Best apProximation

If U is the partial isometry from HI to H2 as given in Proposition 4.17, then T = U A has the Schmidt decomposition, 00

(423)

Tx = UAx = LUn (x,xn)Yn, n=O

where {Yn}~=o

= {Uxn}~=o

is an orthonormal set in H2 Indeed,

IIUXn !l IIU(xm +xn )1I 2

Ilxnll = 1, IIxm + xnl1 2 = 2,

=

+ ixn )11 2 = I\xm + iXnl12 = 2, imply both Re(Uxm,Ux n ) = 0 and Im(Uxm,Ux n ) = O. Now IIU(xm

(x, T*y)

implies that T*y TXn T*Txn

=

(Tx, y)

= ~ Un (x, Xn) (Yn, y) = ( x, ~ Un (y, Yn) xn)

E:'=o Un (y, Yn) Xn, SO that

=

= =

= UnXn' = U~Xn and TT*Yn = T(unxn} = u~Yn.

UnYn and T*Yn T*(unYn)

Clearly we can extend {Yn}~=o to an orthonormal basis of eigenvectors for TT* since kerU = (rangeT).L = kerT*. Thus we may assume that {Xn}~o and {Yn}~=o are orthonormal bases of eigenfunctions for T*T and TT* respectively.

Now suppose the eigenvalues {un}~=o are arranged in nonincreasing order. Then the operator 00

ANX

=

L

Un (x, Xn) Xn

n=N

satisfies IIANII = UN, for all N ~ 0, and since N-l

T - UA N =

L

Un (X,Xn)Yn

n=O

has rank N, we easily see that sN(T)

=

IIUANII

=

IIANII

= UN,

which completes the proof of (4.20). Example 4.18 The Schmidt decomposition (4.23) corresponding to an m x m matrix T can bear little resemblance to its Jordan canonical form. For example, if T is the Jordan block ),[ + N where N is the nilpotent matrix

0

1

0

0

0 N=

0

0 1

0

0

0

0

then TOT

= (XI + ~)(>.I + N) = IAI2 I + ANt + XN + J

74

4. Toeplitz and Hankel Operators

where I = Nt N is the identity matrix I but with 0 rather than 1 as the entry in the top left corner. The eigenvalues of T*T are a complicated function of A and m. The space gp defined in (4.18) is a Banach space when 1 ::; p < 00. We let goo denote the space of compact operators. When H = HI = H2, the space of Schatten class operators g2 consists precisely of the Hilbert-Schmidt operators T on H, namely those satisfying 00

00

00

00

n=O

n=O

n=O

> IITII~s ==

2: IITvnll~ = 2: (Tvn, Tvn) = 2: (vn' T*Tvn) ,

where {vn}~=o is any orthonormal basis of H, and where the sum is easily seen to be independent of such a basis. Indeed, Theorem 4.16 shows there is an orthonormal basis of eigenvectors {wn}~=o of the positive self-adjoint compact operator T*T, i e. T*Twn = 'TnWn where 'Tn ~ 0 are the eigenvalues ofT*T. Then 2:~o !(vn ,Wj)!2 = IIWj 1I~ = 1 for all j and

~ {V., TOTvn) ~ ~ (t, {Vn, Wi) Wi' ~ {Vn, W;) T'Tw; )

(4.24)

~(~(Vn'Wi)Wi'~{Vn'Wj)'TjWJ)

=

00

00

00

2:L!{vn,Wj}!2'TJ = L'Tj· n=Oj=O j=O As a result, the characterization of compact Hankel operators H

H
= p_(zn L

(p(j)zj)

=

jEZ

L (P(t - n)z£ i=-oo

and 00

IIH
= IIH
(H
(4.25)

n=O 00

=

-1

-1

L L !(P(t - n)!2 = L jt! 1{P(t)1 2 n=O£=-oo £=-00 2

= £=-00

V(II»

Thus H

Theorem 4.19 Let 1 ::; p < .1. P_

00

and
E VMO.

Then H
E

gp if and only if

Here is the theorem of Adamyan, Arov and Krein which says that to compute the s-numbers of a compact Hankel operator, we need only take the infimum in (4.19) over finite rank Hankel operators. This seems to suggest that there might be a bounded projection onto the closed subspace of bounded Hankel operators

75

4.3. Best approximation

that does not increase the rank of an operator. But in fact there is no bounded projection at all - see [29]. For an alternative presentation of the material here see the excellent volumes [30] and [31] by Nikolski. TheoreIIl 4.20 If A: H2

-+ H~

is a compact Hankel operator, then

= min{IIA n = 0 of (4.26)

sn(A)

Proof: The case Hankel operator A we have

Hcpll : rank(Hcp) ~ n}.

is so(A)

=

(4.26)

IAI =

SUP>'EO'(A)

IIAII. For any (4.27)

Now choose a natural number n E N such that Sn-1 (A) > sn(A) = ... = Sn+k-1 (A) > sn+k(A),

where k is the multiplicity ofthe s-number sn(A). Clearly it suffices to prove (4.26) for such n. Set S = Sn and define E+ E_

{~ E H2 : A* A~

= {7] E

= S2~},

H~ : AA*7]

= S27]}.

Then

= dimE_ = k,

dimE+

AE+

= E_,

A*E_

= E+.

We say that a pair (~, 7]) E H2 X H~ is a Schmidt pair for A corresponding to s if A~ = S7] and A*7] = s~. We let SeA; s) denote the set of Schmidt pairs for A corresponding to s. Note that E+ = {~ : (~,7]) E S(A;s)} and E_ = {7]: (~,7]) E S(A;s)}. We now claim that ~1~2 = '1117]2 if(~l' 7]1), (~2' 7]2) E SeA; s).

(4.28)

Indeed, we have ~1~2,7]17]2 E L1(11') and for n 2: 0, (4.27) yields

~1~2(-n)

(Zn~1'';2) = ~s (zn';1,A*7]2) = ~s (A(Zn~1),7]2) ~ (p_(zn A';1),7]2) = (Zn7]1,7]2) = M2(-n), s

and similarly, ~(n) = M2(n) for n 2: 1. This completes the proof of (4.28). Thus the function

(4.29)

First we observe that if (';,7]) E SeA; s), then (~, 7]) E SeAs; s): indeed, As';

Next, As~

=

= sP_q.;) = s7] and A:7] = sP+(~7]) = s(.

A'; if (~, 7]) E SeA; s). Now from (4.27) it follows that n 2:

o.

The closed subspace S generated by {zn{ : n 2: 0,'; E E+} is invariant under multiplication by z and hence S = OH2 for some inner function 0 by Beurling's

4. Toeplitz and Hankel Operators

76

Theorem C.16 (p. 193). Denote by 8 the operator of multiplication by () on H2 Clearly A8 = A s 8. We now claim that m

== dim(H 2 e ()H2)

~ n.

Lemma 4.21 The multiplicity of the s-number s of the operator A8 is at least k+m. Proof: First we show that

(A s 8)*(A s 8)'F,; = s2'F,; for ~ E E+ and any divisor T of (), i.e. T is an inner function with T- I () E H oo . For this we note A*(zf) = zA*(1) by (4.27), and hence the transformation Jf == zf maps E+ onto E_ Since () is a divisor in H2 of all the functions in E+, we see that 7J is a divisor in H:' of all the functions in E_. Let,; E E+ and 'fJ = sA~ E E_, and 'fJ = Op, where p, E H:'. Then

(A s 8)*(A s 8)'F,;

= (As8)*sP_(~()'F~) s(A.8)*(p,'F)

= s(A s 8r P_(p,'F)

= S2 P+(~Op,'F) = s2'F~. 'fJ

Now choose divisors ()l,' , ()m of () so that ()m = () and ()i+1();l E H oo with not constant for 1 ~ i ~ m - 1, and ()1 not constant Then, as we have shown, the subspace E j == span{E+, ()1E+, .. ,OjE+} consists of the eigenvectors of the operator (A s 8)*(A s 8) corresponding to the eigenvalue s2. Clearly Ej+1 \ E j -I cp, and from this it follows that ()i+l();l

dimker«A s 8)*(A s 8) - s2I):::: dim Em :::: k

+ m,

where k is the multiplicity of sn(A). This completes the proof of the lemma. Now we return to proving m ~ n From this and the lemma we obtain

Sn+k(A) < sn+k-I(A)

We have sj(A8) ~ sJ(A)

11811 = sJ(A).

= .. = sn(A) = sm+k-l(A8) S sm+k-l(A)

Thus m + k - 1 < n + k and m S n. Thus we have proved that dime H2 e () H2) S n, and consequently the operators A and As coincide on a subspace of codimension n. But this means that rank (A - As) S n, i.e (4.29) holds and the proof of Theorem 4.20 is complete. We now recall the following easy best approximation result using duality. Proposition 4.22 If f E £00(1I'), then there exists 90 E Hoo such that

Ilf-goli oo

=

distu>o(j,H OO )== inf Ilf-gil gEH=

00

(4.30)

SUP{12~ 127r fFd()/. FE Hd, IIFIII s I} ~ Ilflloo Moreover, if there exists Fo E HJ, lIFo II 1 ~ 1, with 2~ J~7r f Fod() = distLoo (j, HOO), then the extremal go E H co is unique and If(e i9 )-9o(e i9 )1 =distLoo(j,H OO ) ,

a.e. (JE1I'.

(4.3])

4.3. Best approximation

77

Proof: The first equality in (3.24) is

= inf{lIx* + kll : k E y.L}, and with X = L1, Y = HJ, X* = L'XJ and y.L = Hoo, we obtain the final equality sup{l(y,x*)1 : y E Y,

!\y!!

S; I}

in (4.30). A normal families argument shows that a best approximation go E Hoo exists as in the first equality of (4.30). If a dual extremal function Fo exists, then

r (f - go)FodO

1

distLoo(f,H OO ) = 21r

Jo

27f

S; IIf -

golloo lI Foll 1 S; distLoo(f, H OO ),

implies that 1

distL=(f, H OO ) = 21r

f27r

Jo

(f - go)FodO = IIf -

golloo IIFo!!l =

IIf - go 1100 ,

and hence

f(e i9 ) - go(ei9 ) distLoo (J, Hoo) Thus the extremal

go

f(e i9 ) - go(ei9 ) Fo(e i8 ) a.e. () E 'JI'. IFo(ei8 ) I' Ilf - golloo is uniquely determined and (4.31) holds.

In general the best approximation is not unique Another simple condition that if> sufficient for uniqueness is the following condition in terms of Hankel operators. The resulting form of the extemal will prove very useful. The proof uses the above duality argument together with the inner-outer factorization of H2 functions. Proposition 4.23 Let

h(z}_
a.e. z = ei9 E 'JI',

(4.32)

where AE C (with IAI = distLoo(
!\H",goll£2 = IlH",!I. Now choose

fo

E Hoo such that

1I

= P_(
=

P_«

IIH",U

llH",goll£2 = IIP-«

and

j

=

IIH
a. e. on 'JI'

Now let H
78

4. Toepllt. and Hankel Operators

a.e. on 'll', it follows that Ihol = IIHl"lIlhll a.e. on 'll' and so constant ~ E C with I~I = IIHI"II. Thus H 1"90 (z )


90(Z)

=

=

-=-ho""'("""z)'--z-=-bo""7(z---) h1(z)b1(z)

he

= Xh 1 for some

hI ( Z) .~----=-~

=

~hl(Z)zbo(Z)bl(Z),

for a.e. z = eifJ E 'll' and where I~I = II HI" II = distu",(cp,Hoo). Finally, if P_cp E VMO('ll') , then Hartman's Theorem, Corollary 4.11 (p. 67), shows that H

BMO(l') = {u+v: u,V E Loo(l')}.

(4.33)

Indeed, if f E BMO(1f), then

f

= u + V = (u + iv) + (v - iv)

= w

+9

where

w

= u+iv E L oo ,

9

=

-i(v + iV) E BMOA('ll').

Using a normal families argument one can show there exists an extremal function 90 E BMOA('ll') such that

IIf -

901100

=

inf

gEBMOA(1J.")

IIf -

91100

= distLoo (I, BMOA(1f».

Furthermore, if f E VMO, then cp = f - 90 E Loo and P_cp = P-f E VMO by Sarason's characterization of V MO in (4.16). Thus Proposition 4.23 implies that hzb f - 90 = cp = cp - fo = ~h

as in (4.32) since fo clearly vanishes in (4.32) for cp = f - 90 with 90 extemal We will also need to characterize invertibility of Toeplitz operators TI'" Lemma 4.25 If T

f

Proof: A calculation shows T; = Tip. Let

E kerTI" and 9 E kerT;. Then

both cpf and rp9 are in H=-, hence both CPf9 and rp9f = CPf9 are in Hi. It follows that CPf9 E Hi n H:" = {OJ. Since cp =f. 0, either f or 9 must vanish on a set of positive measure, hence by the inner/outer factorization Theorem C 14 (p. 190), one of them must vanish identically. It is now immediate that TI" is invertible if and only if it is Fredholm and index(TI") = O. If T

0 such that c IIflb ~ IITI"fll2 ~ II cpf II 2

,

f

E

H2.

Consequently,

c II:zn fll2 = c IIfll2 ~ IIcpfll 2 = IIcpZ" fll2 , and since the closure of u;::"=o:zn H2 = L2, we obtain c IIfll2 :::; IIMl"f1l2, and thus that 1. E L oo . Finally, let h be an outer function satisfying Ihi = Icpr!. Since I"

79

4.3. Best appl"OxiDlation

o
closure(h(lJ))) ' [ < Tp 1 - = T~h h = T;;TepTh is invertible by Wintner's Theorem' [

IITi111 <

A is a bounded Toeplitz operator, hence by Theorem 4.1, its symbol 'i{1-l is bounded Thus 'i{1-1 E L oo n H2 = Hoo. 00,

Thus matters have been reduced to characterizing invertibility of Toeplitz operator:;, Tu with unimodular symbol u. In this direction, we have the following consequence of Nehari's Theorem The operator Tu is left invertible if and only if Tu is injective and has closed range if and only if Tu is bounded below. However, IIfl12 = IIM ufll 2 = IITu fll 2 + IIHuf\l2 and so Tu is bounded below if and only if IIHull < 1. But Nehari's Theorem (4.1) shows that IIHull = distv=(u,HOO('IT'», and so altogether we obtain that Tu is left invertible if and only if dist Loo (u, Hoo ('IT')) < 1. This is the first assertion in the following characterization. Theorem 4.26 Suppose that lui = 1 a e on'IT'. Then. 1. Tu is left invertible if and only if dist £0"" (u, HOO ('IT')) < 1 2. Tu is right invertible if and only if dist Loo (u, H oo ('IT')) < 1. 3 IfTu is invertible, hE Hoo and lIu - hlloo < 1, then h is an outer function. Proof. Assertion 1 has already been proved and 2 follows from 1 using Tu;. To prove assertion 3 we note

III -

T~Thll

T: =

= III - Tuhll = 111 - uhlloo = lIu - hll oo < 1

implies that T:Th is invertible. Since T: is invertible, it follows that Th is invertible on H2, so h- l E H2. Now Lemma 4 25 implies h- l E Loo n H2 = H oo . In addition we need a Fredholm test for Toeplitz operators with symbol in Hoc +C. Theorem 4.27 Let 'P E H oo + C. Then Tep is a Fredholm operator if and only if 'P is invertible in Hoo + C. Proof' From 'P- l E Hoc+C and Hartman's Theorem 4.11 we obtain that Hep-1 is compact, and hence so are I - T oso that 'P;;':ter E Hoo where 'Pouter is the outer function with I'Pouter I = I'PI given by Theorem C.13. Thus u == -;;;-'£E HOO + C and by Hartman's Theorem 4.11 again, Touter Hu is compact. Since u is unimodular we have H;];Hu;Tu = TuH~Hu using (4.34) -1 = T
QC == (HOC

+ C) n (Hoo + C),

4. Toeplitz and Hankel Operators

80

i.e. QC is the largest self-adjoint subalgebra of H oo

+ C.

Lem.ma 4.28 QC = V MO n Loo Proof: We obtain QC = V MO n Loo from cp = P_I{) + P+cp and (j; = i[p-cpP+cp + (j;(O)]. Indeed, I{) E Hoo + C implies P_cp E P_C c V MO, and similarly I{) E Hoo + C implies P+I{) = cp(O) + P-Ci5 E V MO Conversely, if cp E V MO n Loo, then (4.16) shows that there are u, v continuous such that

cp = u

+ V = u + iv -

Then cp bounded implies that P+v we see that Ci5 E H oo + C.

E

2P+v + const.

H oo . It follows that I{)

E

Hoo+C, and similarly

Finally, we need the following connection between Hu and Hu; for unimodular functions u. But first we list some identities involving Toeplitz operators Tip and Hankel operators Hlp on H2, and the multiplication operator Mip from H2 to L2 defined by Mlpf = I{) f For cp, 'I/J E Loo and u unimodular, we have

Mip

T;

(Tip'" - TlpT",) 1- T;;Tu H;;;Hu;T..

Tip + Hlp, T-q;,

(4.34)

H;PH"" H~Hu,

(/ - TuTu;)Tu = Tu(/ - Tu;Tu) =

As a consequence, we have kerTu = kerT';Tu = ker(I -

TuH~Hu

H~Hu).

Corollary 4.29 Let u be a unimodular function in Hoo

+ C such that closure

(TuH2) = H2. Then sk(H;;;) :s; sk+n(Hu), k ~ 0, where n = dimkerTu = dim{J E H2 . H~Huf = J}. Proof: Recall from Proposition 4.17 (p 72) that a bounded operator T on a Hilbert space H has apolar decomposition T = U(T*T)~ where kerT = kerU, and the partially isometric part U of T maps H eker T isometrically onto closure(T( H». From (4.34) we have H;;;Hu;Tu = TuH~Hu, and we now claim that we also have

H;;;Hu;U = U H~Hu, where U is the partially isometric part of Tu Indeed, if A and B are self-adjoint operators on a Hilbert space H with AT = TB, then AU = UB where U is the partially isometric part of T To see this note that T* A = BT* and so

T*T B

= T* AT = BT*T.

Thus B commutes with every function of T*T, in particular with (T*T) ~. Thus AU(T*T)~

= U(T*T)~B = UB(T*T)~,

and if we multiply through by (T*T) ~ on the right we get

AUT*T = UBT*T Thus the operators AU and UB coincide on closure(T*T(H», and we now show that they also coincide on the complement ker T of this space. Since B takes ker T into itself (AT = TB) and kerT = kerU, we see that AUx = 0 = UBx for all x E kerU.

We are now in a position to investigate the following two morally equivalent problems: Problem 4.30 Best approximation by analytic functions in the uniform norm. For I E BMO('JI'), there is by Remark 4.24 a function go E BMOA('JI') such that I - go E Loo('JI') and

III -

go 1100 = inf{111 -

glloo . g E BMOA('JI')}.

The function go (not in general unique) is called a best approximation or extremal for f. What conditions on a space X contained in BMO('JI') are needed in order that I E X implies go E X? (4.35) In the case X = VMO, Proposition 423 (p. 77) with 'P = I - go implies that such a function go is unique since P _ is bounded on V M 0 by Sarason's characterization of VMO in (4.16). Thus in VMO we define the nonlinear approximation operator A' V MO - V MO by AI = go Problem 4.31 Restoration of a unimodular function. Given a space X c BMO('JI') what conditions on a unimodular function u are needed in order that P_u E X implies u E X? (4.36) Remark 4.32 To see the connection between these problems in certain situations, suppose that X c BMO('JI') satisfies P_X c X. Then from Proposition 423 we obtain that I - AI = .xu where .x E C and u is unimodular. Clearly P_u = P_I E X and thus (4.35) holds for I whenever (4.36) holds for u Conversely, assume that u is a unimodular function satisfying in addition TuH2 = H2, where Tu is the Toeplitz operator with symbol u. We then claim that - P+ (zu) = P _ (zu) - ZU is the best approximation for P _ (zu). Indeed, by assumption there is I E H2 such that Tul = 1 Thus Tzul = 0 and by Theorem 4.26, we must have dist£oo(zu, HOO) = 1. This however says precisely that -Pt(zu) is the best approximation for P_(zu). Thus if P_(zu) E X, then (4.35) with I = P_(zu) implies that go = -P+(zu) E X, and so (4.36) holds for the unimodular function zu place of u. 4.3.2 Rational approximation in the BMO norm. The space BMO('JI') comes equipped with the norm

II 'P II BMO(1l')

-

=

inf

II'P - glloo + gEBMO inf 11"1,0 - glloo + 1<;0(0)1 A (1l')

inf

IIP-'P - glloo + qEBMOA(T) inf IIP+'P - glloo + 1<;0(0)1,

gEBMOA(1l') gEBMOA(T)

which satisfies

(4.37) For n 2: 0, let Rn consist of all rational functions with poles not on the circle 'lI' and such that the number of poles (including infinity) counted according to multiplicity is at most n. For IE BMO('JI') we define

rn(f)

== distBMO(1l')(j, Rn)

Definition 4.33 A linear space E of sequences on Z+ is called ideal if {Xn}~=O E E and IYnl ~ IXnl , n 2: 0, imply {Yn}~=o E E.

(4.38)

82

4. Toeplitz and Hankel Operato....

Definition 4.34 A class X of functions contained in BMO admits a description in terms 0/ a rational approximation in the BMO norm (more briefly is an 'R.-space) if there is an ideal space E such that / E

X if and only if {rn(f)}~=o

E

E.

For example, V MO is an 'R.-space if we take

E = CO(Z+) = {{xn};::'=o : n--+O lim Xn = OJ. .1

Another example is Bt with E = fP(Z+), 1 ~ P < 00. This is seen to be an 'R.-space by Peller's Theorem 4 19 (p. 74) together with the formula

cp E BMO,n 2: 0,

(4.39)

which follows in turn from Theorems 4.20 and 4.7 (p. 64). The case p = 2 is especially easy. The equalities (4.25) and (4.24) show that if Pn}~=o are the eigenvalues of H;H

00

1/P-cplI:~ = j/P+~II;(ID) = I!H
2

n=O

Theorem 4.20 shows that

sn(H
sn(H
The same equality holds for P_cp, and now we use

together with (l-az) = 101 -:.z for Z E 11' to obtain that Q-z -QZ '

Remark 4.35 It is also true that Bj (11') c V M 0(11') Here is a sketch of the proof of the analytic case V(D) C V MOA(D). For b E V(D), { 1T(I)

!b'(z)1 2 (1 - Izl2)dz

~ C III

(

Ib'(z)!2 dz

=

o(!Ij),

11-lz/ 2 -5.111

i.e. dlLb(z) = lb'(z) 12 (1 - Izj2)dz is a vanishing H2-Carleson measure. Using the ideas in the proof of Fefferman's Theorem 4.4 (p. 59), we can conclude from this that the mean square oscillation of b is vanishing, i.e. bE VMO('IT'}.

83

4.3. Best approximation

We now show that the best approximation problem is solvable in R-spaces (in 1 particular for Bi (']f)), and provide conditions on a unimodular function u that are sufficient for solving the restoration problem. The following theorems are from Peller and Khrushchev [34]. See also Rochberg [35] for a different and more real variable approach to restoration. Theorem 4.36 Suppose X is an R-space and u is a unimodular function on 11' such that P_ u EX. If the Toeplitz operator Tu has dense range in H2, then P+u E X. Thus u E X if X is also linear. Theorem 4.37 Let X be a linear R-space satisfying zX = X, and let u be a unimodular function on 11' satisfying P _ u EX. If u E V M 0, then u EX. Theorem 4.38 Let X be an R-space. Then the best approximation operator acts in X, i.e. AX c X. Moreover,

f E VMO,n 2: 0, and if in addition P-f -=f 0, then rn(Af)

~

(4.40)

rn+l(f)

...

Corollary 4.39 Let X be either B$ or VMO. 1. If u is a unimodular function in X such that both closure(TuH2) P _ u EX, then u E X and

2. If u is a unimodular function in V M

= H2 and

° such that P_ u EX, then u EX.

3. If f EX, then Af E X and

IIAfli x

4. If fERn, then Af E

~ C

Ilflix . Rn If also P-f = 0, then Af ERn-I·

Proof: (of Theorem 4.36) Since P_u E VMO, we have Hu compact by Hartman's Theorem Corollary 4.11. Corollary 4.29 (p. 80) then implies with m = dim ker T u ,

sn(Hu)

= sn(H;')

~ sn+m(Hu) ~ sn(Hu),

n2:0

Now (4.39) yields

= rn(P_u) = sn(Hu) ::::; sn(Hu) = rn(P_u), P+u E X since X is an R-space. Thus u = P_u+P+u E X

rn(P+u)

which shows that if X is also linear. Proof: (of Theorem 4.37) Since u E VMO n L=, Lemma 4.28 shows that both u and u are in H= + C, hence u is invertible in H= + C. Theorem 4.27 now shows that Tu is a Fredholm operator, and by Lemma 4.25, Tznu is invertible if n = index (Tu) , and in particular has dense range in H2. From the assumptions P_u E X and zX = X we obtain P_(znu) = znp_u E znx = X, and it now follows from Theorem 4.36 that znu EX. Hence u E zn X = X. To prove Theorem 4.38 we will use the following lemma. Lemma 4.40 Let u = b~ where b is an inner function and h is an outer function. Then the Toeplitz operator Tu has dense range in H2.

4.. Toeplltz and

84

Hankal Operato....

Proof: Suppose that I is non-zero in H2 and I ..1 Tu H2 . Then (I, ~) for all 9 E H2. Let I 9 = linnerbh. Then 0=

= linner 1000ter

(I, b~9)

=0

be the inner / outer factorization of I and set

= (jinnerlOfLter, iiIinner ) = (jouter, Ii),

and we have louter(O)h(O) = o. But outer functions cannot vanish in contradiction proves the lemma. Proof: (of Theorem 4.38) Let IE VMO. By Remark 4.24,

II - All

=

]I}

and this

IAI ~ 0

a.e. on 'Jr. If A = 0, then (4.40) holds trivially. If A =I- 0, then by Remark 4.24 again we have that

u == A-1(f - AI)

b~\

=

where b is an inner function and h is an outer function. Note that u is unimodular. By the above lemma, Tu is dense in H2. Furthermore, kerTu =I- {OJ since Tu(hb) = O. Thus m = dim ker Tu ~ 1 and so just as in the proof of Theorem 4.36, Corollary 4.29 and Theorem 4.20 yield

Tk(P+U)

= Tk(P_U) = sk(Hv:) = sk(H;') ::::;

Tk+l(P_U),

k ~ L

Tt(P+U) ::::; Ti+l(P_U) = Ti+l(P-I),

f ~L

sk+m(Hu )::::; Sk+l(Hu )

Since P+AI = AI and P_AI

Tt(P+I - AI)

=

=

= 0 we obtain that

Now fix n ~ O. Let Tn+l(f) = III - 911 BMo where 9 is a rational function in 'R.n+l. We have from (4.37) that

III - 911 BMo

=

IIP+I - P+9I1 BMO

+ IIP-I -

Let k be the number of poles of 9 outside the closed disk and P-9 E R n- k + 1 . From this it follows that

Tk(P+1)

P-9I1 BMo . ]I}

u 'Jr. Then P+9

E

Rk

+ Tn-k+l(P-1) S Tn+l(f).

So altogether we have with f = n - k,

Tn(AI)

< Tn (P+I - AI) + Tn(P+1) S Tn-k(P+I - Af) + Tk(P+f) S Tn-k+l(P-f)

+ Tk(P+f) S Tn+l(f).

Proof: (of Corollary 439) Since X is both an R-space and a Banach space, we combine Theorems 4.36 and 4 38. Remark 4.41 In view of the elementary proof given above that

BJ is an R1

space, we have succeeded in giving a complete proof that A acts boundedly on Bi, i.e. if I E then AI E and

BJ,

Bl

IIAIIIBi ::::; 2

c 11111 B2i·

.... .,. DeIR

appl"OXlmtlflOb: 1

1

Problem 4.42 Recall the space Xi ('JI') c Bi ('ll') consisting of (boundary values of) holomorphic functions h such that d/1-h = IVlPh (z) 12 dxdy is a V-Carleson measure. This space arose in Xiao's proof of the corona theorem for Mv in Subsubsection 3.4. Does the best approximation operator A act boundedly on Xl? An affirmative answer would simplify and unify the approach to corona theorems there.

CHAPTER 5

Hilbert Function Spaces and Nevanlinna-Pick Kernels Recall from Definitions A.20 and A.2l (p. 155) in Appendix A that a kernel function k : n x n ~ e on a set n is positive semidefinite, and that the associated Hilbert function space 1-l,. corresponding to k has kw(z) = k(z, w) as reproducing kernel: If n c en, then 1-l,. consists of holomorphic functions if and only if k(z,w) is holomorphic in z and antiholomorphic in w The Nevanlinna-Pick property for a kernel k is defined in Definition 5.5 (p. 92) (for spaces of holomorphic functions on the disk - the general definition is similar [1], [20]), where it is used to establish that the interpolating sequences for the Hilbert function space 1-l,. and its multiplier algebra M'Hk coincide. The complete Nevanlinna-Pick property for a kernel k is defined in Definition 5.35 (p. 119), where it is used to establish that the baby corona problem for the Hilbert function space 1-l,. is equivalent to the corona problem for its multiplier algebra M'Hk' We will prove that the Hardy space H2(ID» has the Nevanlinna-Pick property by a lengthy but very elegant route - namely by developing the HOC functional calculus and establishing Sarason's characterization of the commutant of the compression T of the shift operator to a subspace K of H2 that is invariant under the backward shift operator.

5.1 The commutant

Given a closed subspace M of a Hilbert space H, we have by Theorem A.IO the orthogonal decomposition H = M EB M.L. Let PM denote the orthogonal projection of H onto M, i. e. PMX = m if x = m + m.L with m E M,m.L E M.L. For a bounded linear operator S on H we define the compression T = PMS 1M of S to the subspace M to be the operator sending x E M to the orthogonal projection PM of Sx, i.e. Tx = PMSX, x E M. Now let S be the forward shift operator on the Hardy space H2(ID», i.e. multiplication by z, and let Me be the closed subspace Me

= H2 e8H2,

where 8 is an inner function in the disk, i.e a bounded holomorphic function on the disk with boundary values of modulus one almost everywhere - see Theorem C.12 (p. 189) for a characterization of inner functions. Note also that by Beuding's Theorem C.16(p. 193) and the discussion preceding it, the spaces Me with 8 inner are precisely the closed subspaces of H2 invariant under the backward shift S*

...,.

5. Hilbert Function Spaces and Nevanlinna-Pick Kernels

88

where S* f(z) = Z(-l fez) - f(O». These are in turn the complex-valued nilpotent model spaces arising in Theorem C.15 (p. 192). 5.1.1 The H oo functional calculus. Let us fix an inner function K = Me. Thus K is invariant for S*. Let

e and write

T=Te =PKS IK

be the compression of S to K. We now ask for a characterization of the commutant of T, the set of bounded operators on K that commute with T. If A = a(T) = ~:=o anrn is a polynomial in T, then A commutes with T and we show below that IIAII ::; Iialloo. Taking uniform limits we can extend this to a in the disk algebra A(ii) However, the operators aCT) for a E A(ii) still do not exhaust all of the bounded operators on K that commute with T. In the hope of obtaining the commutant of T we would like to extend the functional calculus to a E Hoo. However, the operator T is not normal, so we cannot apply the well known LOO(a(T» calculus for a normal operator on a Hilbert space (Theorems D.2 and D.3 (p. 196) in Appendix D). Moreover, the general calculus in a Banach algebra applies only to functions holomorphic in a neighbourhood of a(T) (this calculus uses the first three assertions in Theorem 3.1 (p 35) to establish a theory of holomorphic functions - see Theorems 10.27, 10.28 and 10.29 in [38]), which does not include Hoo since from Theorem 5.4 (p. 91), aCT) = {z ED. 8(z)

= O} U {A E 1r: lim zinf 18(z)1 = OJ. .... ,\

Fortunately, one can develop an Hoo functional calculus for Tusing Beurling's Theorem C 16 and the inner/outer factorization of H2 functions (Theorem C.14) and this will occupy the first part of this subsection (see also the book by Sz.Nagy and Foi8.§ [47]). When this is accomplished, we will have that the commutant of T includes aCT) for a E Hoo. Sarason's Theorem 5.2 below then uses Nehari's Theorem 4.2 (p. 55) to characterize the commutant of T as preci&ely those operators arising from the functional calculus as aCT) for some bounded holomorphic function a, and with equality of norms We end this subsection by applying Carleson's Corona Theorem for two generators to obtain the spectral mapping theorem 5.4 for this calculus. We follow the development in Lecture III of Nikol'skil [29], beginning with the identities T* = S* IK, Tn = PKS n IK, n 2 0, (51) which in turn follow from «s*)nx, y) = (x, sny) = (x, PKsn y ) for x, y E K and n 2 O. Indeed, the case n = 1 gives T* = S* IK and the general case then yields (x, my)

=

«s*)nx, y)

=

(x, PKSny)

Recalling the definition of the greatest common divisor GCD in Corollary C 18 (p. 194), we can now obtain an HOC) calculus for T and T*. Theorem 5.1 Let e be an inner function, K = Me and T = PKS IK. For rp E HOC) define rpt(z) = rp(z) and rp(T) = PKM

5.1. The

COIDIDutant

oo _ B(K) and T* : H= - B(K) given by 'P - 'P(T) and 'P(T*) respectively, are both linear and multiplicative, and satisfy

2. The maps T: H

'P

-+

T*('P) = 'P(T*) = 'Pt(T)*. 3. If 1/J

= CeD(8, 'Pinner),

(5.2)

then

8

ker'P(T) = 'lj;H2 88H 2 and ker(l(T*) = H2 8'1j;H 2 ,

so that in particular, 'P(T) = 0 iff 'Pt(T*) = 0 iff 'P E 8H=. 4. li'lnn-+=Tnf = 0 and limn--,>oo(T*)nf = 0 in the norm topology of K for each f E K.

Proof: Assertion 1 follows from IIPKII, IIP+II S 1 and IIMcpl1 = 1I'Plloo. We have P+(Ij5K) c K for 'P E H oo since with j = 8g E 8H 2 = H2 8 K and h E K,

o.

(I, P+lj5h) = ('Pf, h) = (8'Pg, h) =

Thus the maps T and T* are into 8(K) and linear The projection P K from £2 onto the subspace K = H2 88H2 is easily seen to be given by PK

= P+ - 8P+8.

(5.3)

For f E H2 and 'P, 'Ij; E Hoo, we thus have PK'PPK'Ij;f = PK'P{P+'Ij;j - 8P+8'1j;J} = PK'P'Ij;f,

since P + 'Ij; j = 'Ij; j and 'P8 P + 8'1j; f E K.l., and this establishes the multiplicative property of T Finally for j, 9 E K,

and (I,cpCT*)g)

=

(J,P+lj5tg)

=

(J,lj5tg)

=


since Ij5t = 'Pt, and this establishes (52). From this and the multiplicativity of T, we also obtain the multiplicativity of T*, and this completes the proof of assertion 2.

By definition ker'P(T) = {J E K· 'Pf E K.l. = 8H2}.

*

However, 'Pi E 8H 2 if and only if 8 divides 'PinnerJ;nner. Since and CP''1t er are relatively prime (no non constant inner function divides both), this last condition is equivalent to the condition that J;nner is divisible by This in turn gives

*

ker'P(T)

=

K

8 8 8

n 'Ij; H2 = (H2 8 8H 2 ) n 'Ij; H2 = 'Ij; H2 88H2.

Similarly we have kercpt(T*)

Since P+(Ij5K)

= kercpCT)* = {J E

K: PKlj5j

= a}.

c K (see above) we obtain

ker'Pt(T*)

= =

K : P+'l5j = o} = {j E K : j ..l 'PH2} (H2 8 8H 2 ) n (H2 8 'Pinner H2 ) = H2 81/JH2,

{J

E

90

5. Hilbert Function Spaces and Nevanlinna-Pick KerneJ..

where we have used that CPouterH2 = H2 by Corollary C.19 (p. 194). This completes the proof of assertion 3. Finally, for all n ;::: 0 and f E K,

II(T*)nfll~

=

lI(s*)nfll~

2

00

=

L

Ii
I

-->

0

k=n as n

--> 00.

Using (5.3) we have

IIPKznfll~

= IIzn f -

8P+8fznll~ = llzn 8f - p+8fznll~ =

ll(I - p+)8fznll~,

which yields the corresponding statement for T. This completes the proof of assertion 4 and hence also the proof of Theorem 5.1. Now we can give Sarason's characterization of the commutant of T.

Theorem 5.2 Let e be an inner function, K = H2 e eH 2, T = PKS IK where S is the shift operator on H2, and suppose that A is a bounded operator on K that commutes with T, i.e. AT = T A. Then there is a E Hoo that interpolates A in the sense that A = aCT) and IIAII = IIalioo Proof: Consider the operator A* : H2

A*

-->

H:' given by

= MeAPK ,

which clearly satisfies IIA* II = li A". We first note that AT = T A is equivalent to PKzAPK = APKz, which by (5.3) is in turn equivalent to

A*z

-

MeAPKz = MePKzAPK

=

Me(P+ - ep+8)zAPK = (Me - P+Me)zAPK (1 - P+)zMeAPK

=

which is the commutation characterization that A. is a Hankel operator. It then follows from Nehari's Theorem 4.2 (p. 55) that A. = H f for some f E Loo where "Hfli = distLoo(f, Hoo) For every h E H2 we have P K (8h) = 0 and so also

P-(f8h)

= Hf(8h) = MeAPK(eh) = O.

Thus Moreover, for 9 E K we have

MeAg

= P-MeAg =

A*g

= Hfg = P-fg

and so by (5.3) again,

Ag

=

8P_(8cpg) = 8(1 - P+)(8cpg)

(5.4)

(P+ - ep+8)(cpg) = PK(CPg) = cp(T)g. Clearly we can replace the function cp E Hoo in (5.4) by any function in the coset cp + 8Hoo , and in particular we can choose a sequence {CPn}~=l in cp + eHoo such that

3 • .1. ".l"ne

commutant;

We may also assume by Corollary A.45 (p. 167) in Appendix A that {
= (
(Af,g)

.... --+00

and we conclude that aCT) = A and llall oo =

IlAII

As an application of Carleson's Corona Theorem for two generators, we conclude this subsection with the spectral mapping theorem for the H oo calculus introduced above. We first recall the definition of the spectrum and point spectrum of a linear operator.

Definition 5.3 Let T be a bounded linear operator on a Hilbert space H. The spectrum u(T) of T is defined by

u(T) = C \

P. E C : (M -

T) is invertible on H}.

The point spectrum up(T) of T is the subset of u(T) given by

up(T) = {>. E C : kerCH - T)

-I {O}}.

Here is the spectral mapping theorem.

Theorem 5.4 Let 8 be an inner function, K = Me and T = PKS IK. For

' E C: inf(!8(z}! + !.1) = O}. zED

(5.5)

Proof: The difficult containment C follows from Carleson's Corona Theorem for two generators together with the Hoo functional calculus. Indeed, if (" is not in the right side of (5.5), i.e.

inf(18(z)1 + ! 0,

zED

then Criterion 35 (p. 39) with N = 2 generators shows that there are bounded holomorphic functions u and v in the disk so that

u(z)8(z) + v(z)(
zE

lJ)

Then assertion 2 of Theorem 5.1 shows that

u(T)8(T) + v(T) (t- u(. E lJ) then k>.(z} = ~1~ is the reproducing kernel for >. and we have l->.z

(f, t/J(>')k>.

>

H2

= f(>.)'I/J(>') =

(ft/J, k>.) H2

=

(I, P+1j)k>.) £2 '

which implies

P+1j)k>. = t/J(>')k>.. Suppose that ( is in the right side of (5.5) Now with 8(>')k>., and then with t/J =

t/J = 8, we obtain P+8k>.

P+(~){k>. - 8P+8k>.}

(ep(T)* - (I)Pe k>. =

(ep(>.) - ()k>. - 8(>')P+(

..

:;=

92

5. Hilbert Function Spaces and Nevanlinna-Plck Kernels

From p+ek>. = e('\')k>. again, we have

IIPek>.lI~

=

Ijk>.l1~ -lIep+ek>.lI~

=

IIk>.lI~ _le(.\.)j2I1k>.II~,

and so we obtain inf 11(
IIflb=l

. f 1I(.1I2

<

III

IIPek>'1I2 inf /. II + le(.\.)III

.1I

>'EJ[)

<

IIk>.lI~ _18(.\.)12I1k>.lI~

>'ED

inf l'EJ[)

(I + 18(.\.)111'1' -

VI _18(.\.)12

(lice

= O.

This shows that
S J?

(5.6)

There is an easy necessary condition for the data in termb of a certain matrix being positive semidefinite If 11M"" II == II'PIIMH S 1 then IIM~/I S 1 and for every choice of scalars {AJ}f=l C C we have 2

2

J

M;(L .\.jk

zj )

j=l

J

=

L

(1 - ~jf,m)kz1 (zm).\.J.\.m,

j,m=l

which is (5.7) We say that the Hilbert space H (more precisely the inner product of H) has thE' Nevanlinna-Pick property if the implication above can be reversed Definition 5.5 The Hilbert space H has the Nevanlinna-Pick property if whenever (5.7) holds, there ib

There is a surprising consequence of the Nevanlinna-Pick property for certain extremal problems. Let Z = {ZJ}~l and Zo 1. Z. To make the following argument rigorous, we may take Z finite and then pasb to a limit Let 10 be the unique solution to the extremal problem Re/o(zo) = {Re/(zo) : I(zj) = 0 for 1 S j <

00

and 1!l1I S I}.

(5.8)

Note that the solution exists and ib unique because for each real t, the element of minimal norm in the closed convex set

E t = {J

E

H . Re/(zo)

= t,/(zj) = 0 for

1S J <

00

and 11111 S I}

93

5.1. The CODlInut"":t

is unique by Theorem A.9 (p. 150). From the definition of fo we have

JAofo(zo}1 = which in terms of the data {o as

(~AJkZ3' fO) ~ ~ Ajkzj

,

= 'rr1~)ll' and {J = 0 for 1 ~ j < 00 can be rewritten

2 00

o~

00

L

L

(1 - ~j{m)kz3 (Zm)AJAm j,m=O Since H has the Nevanlinna-Pick property, there is CPo E MH with norm at most one ~atisfying - JAofo(zo)J2 =

Ajkzj

J=O

( ) = {o = I/o(zo)J \lkzo\l

-Po Zo

and -Po(zJ)

= 0 for 1 ~ J < 00

Thus the function p( z) == CPo(z) OZO (zil satisfies kzo

\lpJI = l cpo II~:~II II

~ IIM~II II II~::\I II ~ 1,

and

and p( zJ) = 0 for 1 ~ J < 00 By the uniqueness of the solution to the extremal problem (58), we obtain the remarkable formula,

10

( ) Z

= CPo

( ) kzo(z) Z

(5.9)

IIkzo JJ·

From the above, we see that every zero set of a function in H is included in a zero set of a function in MH. Indeed, if Z = {Zj}~l is the zero set of IE H, then the extremal problem (5.8) hah a solution provided Zo rt Z But then CPo E MH vanishe~ on Z as well Similarly, we see that ewry int.erpolating set Z for H if:> also an interpolating set for MH Indeed. if Z is interpolating for H, then {k zj }~1 if> a Riesz basis, and consequently satisfies the unconditional basic sequence condition: if laj J~ Jbj J, then 00

~ C lI{aj}~1Ile2(l'z) ~ C lI{bJ}~11Ie2(l'z) ~ C Lbjkz3 j=l

Now we seek to solve the interpolation

cp(zJ) = {j,

1~J

< 00,

with cP E MH of norm at most one whenever I){~j}~dl ~ 8, with (j > 0 sufficiently small But for 8 ~ we have l~jAJ ~ and the unconditional basic sequence condition implies

I %1,

b

2

2 00

=

L

J,m=l

(1 - {j{m)kzj (Zm)AJAm.

94

5. Hilbert; Function Spaces and NevanUnna-Plck Kernels

The Nevanlinna-Pick property now yields the desired solution

J-tz =

L

l!k Zj

11- 2 8

zJ ,

j=l

is interpolating for H, i.e. R. H ----> f2(J-tz) is bounded and onto, if and only zf Z is interpolating for M H, i. e R· H ----> fOO (Z) is onto Now we apply Sarason's Theorem 5.2 to prove that the Hardy space H2 has the Nevanlinna-Pick property. We should emphasize that we do not need the full force of Theorem 5 2 for this application, but only the special case where the inner function 8 is a finite Blaschke product, a case which is relatively easy. For a direct elementary proof of the Nevanlinna-Pick property for H2 see Theorem 2.2 in Garnett's book [20] For convenience we restate the Nevanlinna-Pick property for the Hardy space the unit disk JI)). Definition 5.7 The Hardy space H2 on the disk has the Nevanlmna-Pick property if for every pair of finite sets {Zn}~=l and {Wn}~=l of points in JI)), there exists ip of norm at most one in the multiplier algebra HOO of H2 that interpolates the data, 1

~ n ~

N,

(5.10)

if and only if the matrix

(511) is positive semidefinite. Theorem 5.8 The Hardy space H2 has the Nevanlmna-pzck property Proof. Let 8 be the Blaschke product corresponding to Z == {Zn}~=l' let K be the corresponding model space invariant under S*, and let T be the compression of the forward shift operator S to K. N

8(z) K T

=

II Zn -=-:

Iznl, n=l 1 - Zn Z Zn

Z

E JI)),

H2 e8H 2, PKSIK.

Note that T* = S* IK. Now K is N-dimensional and all the eigenvalues of the finite dimensional operator T on K are simple Indeed, 8H 2 is the space of functions in H2 that vanish on Z, which is clearly the orthogonal complement of the span V{kz"}~=l of the reproducing kernels kz,,«() = l-~n.(" Thus

K = V{kz,J~=l.

95

5.1. The commutant

Now for cp E Hoc = MH2 we have that the adjoint M~ of the multiplication operator Mcp on H2 has eigenfunctions kz with eigenvalues cp(z) since

(!,M~kz)

= (M",J,kz ) = cp(z)J(z) = (J,CP(z)kz ).

Then the compression PKM~ IK of with eigenvalues cp(zn)·

M~

to K will also have eigenfunctions k zn

P K M~ IK kZn

= PKcp(zn)k zn = cp(zn)k zn · (5 12) In particular T* = S* \K= PKS* IK is the compression of the backward shift S* and satisfies T*kzn = znkzn, 1 ~ n ~ N Thus we see that T* has N distinct eigenvalues {Zn};;'=l with corresponding eigenvectors {kz" };;'=1. Now the map A : K ~ K defined by Ak zn = wnkzn commutes with T* (see the remark below) Then A* commutes with T and by Sarason's Theorem,

= cp(T) = cp(PKS IK) = PKM", IK, H oo with IIA*II = Ilcplloo Then A = PKM~ IK and using (5.12) we A*

for some cp E see that (510) has a solution "All ~ 1 if and only if

cp E Hoo

with

\lcp'' ""

~ 1 if and only if

IIAII

~ L But

(5.13) for all sequence& {~n}~=l. The left side of (5 13) is

and the right side of (5 13) is

(t,en k." ,t,'n k.. )~ ,t/ (k."k,)e, ~ Jt,', j

1-

~z,'"

so that (5.13) is equivalent to N

L

1

_

~j~e(1 -

WjWp)

j,£=l

1 -

~ 0,

zJz£

which is the positivity of the matrix (5.11)

Remark 5.9 An operator A : K ~ K commutes with T* if and only if k zn is an eigenvector of A for all 1 ~ n ~ N. Indeed, if T* A = AT* then T*(AkzJ = A(T* k zn ) = znAkzn shows that Ak zn is an eigenvector of T* with eigenvalue Zn, hence Ak zn is a multiple of kzn · Conversely if Akz" = Ankzn' then T* Ak zn = Anznkzn = AT* kzn ·

96

5. Hilbert Function Spaces and NevanUnna-Pick Kernels

5.1.3 Generalized Blaschke products in Hilbert spaces with the NP property. The purpose of this subsubsection is to obtain a purely Hilbert space

proof of Carleson's Interpolation Theorem 2.4 (p. 15) - see Conclusion 5.17 (p. 102) - and a generalization. This will lead us to interpolation in Hilbert function spaces with the complete Nevanlinna-Pick property, and also to generalized Blaschke products in such spaces - a possibly new topic motivated by work of Shapiro and Shields on extremal problems [44]. Let H be a Hilbert space of analytic functions with the Nevanlinna-Pick property Let Z = {zo, Zl, . ,zN-d be an N-point set in the extremal problem (58) above, which we now quickly review. Let 10 be the unique solution to Re/o(zo) = sup{Re/(zo) I(zd = 0 and II/IIH :::; I}. In terms of the data

2

N-l

0:::;

L

=

~o

N-l

-!>.0/0(zo)!2 =

>'Jkzj

j=O

L

(1 - ~J~m)kzj (zm)>'J>'m.

(514)

Jm=O

H

Since H has the Nevanlinna-Pick property, there is
Thus the function p(z)

== of'o(z) 1!~o{iJ) satisfies IIpliH :::; 1, Rep(zo) zo

=

!lo(zo)! and

JI

p(zn) = 0 for 1 :::; n :::; N - 1 By the uniqueness of the bolution to the extremal problem (5.8), we obtain (515) Now we compute the extremal functionb 10 and ..po explicitly. following the approach of Shapiro and Shields in the case of the Dirichlet space [44] Let M = Re/o(zo) = !/o(zo)! Then from (514), M is the largest nonnegative number such that

o :::; [ >'0 for all >'0, ,>'N-I, i e. such that the matrix above is positive semidefinite. Now all of the principle submatrices of this matrix, not containing the upper left entry, have nonnegative determinant Thus M solves

o=

det B - M2 det A

O.l.. Tile

commutant

where B

A

and so we obtain kZ(l (zo) det A - KZoco[Al' Kzo

M

detA

(5.16)

-,

1 _ Kzoco[A]' Kzo kzo (zo) det A

where Kzo = [kzo(Zl) .. kzo(ZN-l)] and colA]' denotes the transposed cofactor matrix of the square matrix [AJ. Next we note that if Z ib a set of points and Hz = {h E H . h(z) = 0, Z E Z} is the closed subspace of functions vanishing on Z, then V{kZ}ZEZ = H~.

Indeed, (h, kz) projection

= h(z) = 0 for

all

Z

E

Z if and only if h E H It follows that the

PV{k'}ZEZI

= PHtl

takes the same values on Z as I, and moreover PH1-f minimizes the norm over those z functions in H taking the same values on Z as I. Thus 10 E V{kz}zEZ and we see that there are numbers AO,· ., AN-l such that lo(z) = Aokzo(z)+' +)..N-1k zN _1 (z). 'vVe mUbt have 1

=

(which follows automatically from the calculations below) and

Thub M M = d et1 B CO [)' B Mel = --co[BJ'el =-r det B det B '

5. Hilbert Function Spaces and Nevanllnna_Plck Kernels

98

where

r

~o

= [

] is the first column in the transposed cofactor matrix co[B]',

IN-l

and we obtain (517)

fo(z)

Since fo(z) =

II kzo(lf k ZQ


H

formula for the multiplier

IIk zo ll H kzo(z)

M detB

{

~

( )}

(5.18)

~ 'nkz .. z

IlkzollH kzu(z)JdetAdetB

{I:\ n=O

k

n

z"

(Z)}

IIk z IIH det[B(z)] kzo(z)Jdet Adet B (J

kz()(z) where [B(z)] is the matrix [B] with its fir&t column replaced by [

.

1

. It

kZ"_'(Z)

= det B. = F"lkMI 80 We also have the formula

is evident from this that det[B(zn)] = 0 for 1 ::; n ::; N - 1 and det[B(zo}]

Thus we have as expected that <po(zn)


since 10 = det A.

II"'Z{) 1111

5.1. The cOIllmutant

99

We now concentrate on the two-point case N = 2. Using the identity (2.26) we have the formulas,

kzo(zo)k z1 (zd

M

-Ikzl (ZO)12

kZ1(Zl)

and


(1

!kZ1 (zo)1 2 - kzo (zo)k Z1 (Zl)

d(ZO,Zl)-l

)-~ (1

-

kzo(zdkzl(Z)) kZl (zdkzo (z)

(1- kzo(Zl)kz1(Z)). kZl (zl)kzo(z)

Summarizing, we have used the Nevanlinna-Pick property to show that there is a unique multiplier 1/1 = 1/1~~ =
2 and 1/1(z ) = 0

IIkzoll211kzJ2

1,

and moreover, we have the explicit formula obtained on page 31 of [43], (

1- l(kzo,kzJI2

IIkzoll211kzll12

d(ZO,Zl)-l

)-t (1-

(kzo,kzJkzl(Z)) (kzpkzJkzo(z)

(5.20)

(1- (kzo,kzJ kZ1(Z)). (k Zll kzJ kzo (z)

We will refer to 1/1~~ as the generalized Blaschke function associated to the pair of points (zo, Zl). It vanishes at Zl and is positive at ZOo More generally, for Z = {zn};'=l' we will refer to the infinite product 00

B~(z) =

II 1/1~~(z),

(5.21)

n=l

as the generalized Blaschke product in MH associated to the set Z = {Zn}~=l with pole at Zo ¢. Z. Suppose now that the measure p'z = 'E~=l IIkz" 11- 2 8z" associated to Z is a finite measure. Then we have

f: n=l

Ikzo~zn)12

2

IIkzo I Ilkz" II

=

f

!kzo (z)!2 dp.z(z) = C zo ·

It now follows from this and the right hand inequality in (C.3) that

B~O(zO)2 = IT 1/J:~(zO)2 = IT n=l

n=l

(1- II Ikzo~Zn)12 2) = IT II IIkz" II kzo

n=l

d(ZO,Zn)2 >

o.

100

5. Hilbert Function Spaces and Nevanlinna-Pick Kernels

Thus we see that the generalized Blaschke product Bzo (z) is not identically zero, has norm at most one in the multiplier space MH, vanishes on Z, and is positive at Zo In fact, ubing the left hand inequality in (C.3), this argument can be reversed and yields the following characterization of nontrivial generalized Blaschke pIOducts Proposition 5.10 Suppose H is a Hilbert space of analytic junctions with a complete Nevanlinna-Pick reproducing kernel k(x, y), so that H = 1tk Fix a sequence Z = {Zj}~l and Zo ¢ Z Then B~(z) is not identically zero if and only if 00

n=l

if and only if f.tz

=

2:;:ll1kzi

11-

2

8 z } is a finite measure.

In particular, Propo&ition 5 10 gives a sufficient condition for Z to be a zero set for H (i e there is f E H with fez) = 0 if and only if Z E Z) IIf.tzll < 00 Thi& recovers some results of Shapiro and Shields when H is the Dirichlet space D(]]]) or a generalized Dirichlet space D.p (see [44] where condition (12) there on .p &hows that D

We can also characterize separation and Carleson embedding fO! sequences Z using J Ikz", (z)j2df.tzC z ) ~ IIf.tzllCarieson fO! all m ~ 1 Proposition 5.12 Suppose H zs a Hilbert space of analytic functwns with a complete Nevanlinna-Pick reproducing kernel k(:r,y). so that H = 1tk Then a sequence Z = {ZJ}~l is separated ,:i::'i,"~k!:;I, zs a Carleson measure for H if and only if

inf Bi\n{

m>l -

Z'm

}(zm)2

== rn>l inf -

~

1- c. and f.tz

= 2:;1 IIkzj

r 28

zJ

II d(zmozn)2 > 0 n¥l

At this point we collect what can be said about interpolating sequences for a Hilbert space H with the NP property Theorem 5.13 Suppose H ttl a Hilbert space of anal1ltic functions with a complete Nevanlinna-Pick reproducing kernel k(x, y) so that H = 1tk Suppose Z = {zJ}~l is a sequence and let f.1z = 2:;:1 IIkZJ W-.2 8 zj be its associated positwe measure. Denote by R the restnction map Rf = {f(Z)}~lo and let Bi\{zm} be the associated generalized Blaschke products Then the relations

1 {=} (2

+ 3) and if Z is l,eparated 3 {=} 4

hold among the following four conditwns 1. 2. 3. 4.

R maps MH onto COO(Z), R maps H onto C2 (f.tz), R maps H into C2 (f.tz), inf7n~l B~\{zm}(Z7n)2 == 8z > O.

0 • .1.

Tlle cOmmutant

Proof. Conditions 2+3 are equivalent to condition 1 by Theorem 5.6, and condition 3 is equivalent to condition 4 by Proposition 5.12

at

Remark 5.14 Let kz" and let

= II~:~ II

be the normalized reproducing kernel with pole

Zn,

B(n)(z) = B~\{Zn}(z) be the generalized Blas('hke product associated to the set Zn = Z \ {zn} with pole at Zn In order to prove that condition 4 implies condition 2, it suffices to show tha.t the Grammian [(9m,971»);;','.n=1 is bounded on £2 where n2':l.

Indeed, =

B(n)(zn)-l (kzm' MB(n)k z,,)

=

B(n)(zn)-l (M'i:J(n)k z"", k zn )

=

B(n)(z17£) ( - - ) 17£ B(n)(zn) kZm' k zn = bn ·

If the Gram matrix [(917£.9n)];;'.n=1 were bounded on ~

=

£2,

we would then have for

{~j}~1 E £2,

sup

If

111111,2=1 n=1

en77nl =

f bUP_1 (f: bUP_1

11111112-1

111111,2-1

< where C*

==

11'1

n=1

m=1

If

sup 1111ll t 2=1 rn,n=1

b:e 77 17£

n

l

(kzm,9n)em77nl ern k Zm ,

f:

n=l

77n9n )

I

C'II~, (mk,·l '

sUPIITIII,2=1 IIE:'=11Jn9nIlH coincides with the norm of the Grammian

[(gm,gn)l~,n=l

on £2 The inequa.lity 1I~lIt2 ::; C* !IL:=1 (17£kzml!H implies condition 2 by standard functional analysih - Corollary A.39 (p 164) in Appendix A Note that using bz ::; B(n)(zn) ::; 1 and PropObition 5 12, we at least have that the functions 9n are uniformly bounded in H'

5. Hilbert Function Spaces and Nevanlinna-Pick KernelS

Problem 5.15 With notation as in the above remark, when is it true that 1IL.:~=l17ngnIlH ~ CIl17I1£2, or equivalently that the Grammian 00

[I MB(rn)C MB(n)k:;,)] B(m)(zm) 'B(n)(zn) m,n=l 00

[(gm,gn)]m,n=l =

\

is bounded on £2? Later in Subsubsection 5.4 (p. 112), we will give Boo's proof [12] that condition 3 implies condition 2, and hence is equivalent to condition 1, for Hilbert spaces H as above having the additional property that whenever a Gram matrix with entries ( k Zi , k zj ) is bounded on £2, then the corresponding Gram matrix with entries / (kzi' kZj ) / is also bounded on £2

Remark 5.16 It is not true that 2 implies 3 in general. The classical Dirichlet space satisfies the complete NP property, and yet condition 2 holds there for a much wider class of ~equences Z than does 3 (see e g. [10], [11) and [7)). Nevertheless, for the classical Hardy space H2, condition 2 is equivalent to condition 3 by Theorem 2.4, and this can in fact be seen directly using Hilbert space techniques since H2 has the Nevanlinna-Pick property. Indeed, the third identity in (6.19) below easily shows that the generalized Blaschke function 'I/J~~ introduced above is given by the usual single product Blaschke function,

'l/JZO(z) = Zl

Z1 -

Z

( ...!l...::EL) 1-Z"lzo

1-Z1 z/z1- z o/ 1-z1 Z o

that vanishes at Zl and is positive at Zoo The open mapping theorem, together with the fact that we can factor out the generalized Blaschke product B~(z) (which is the usual Blaschke product here) corresponding to the zeroes of an H2 function without increasing the H2 norm of the function, shows that 2 implies 4, which is equivalent to 3. For the converse implication 3 implies 2, we give a purely Hilbert space proof in Remark 5.34 in Subsubsection 5.4 below

Conclusion 5.17 Thus for the classical Hardy space H2 we have, given Remark 5.34, a purely Hilbert space proof (in the presence of separation) of the equivalence of the four conditions in Theorem 5.13 (these are conditions 1, 2, ::I and 5 in Theorem 2.4) Problem 5.18 For which Hilbert spaces with a complete Nevanlinna-Pick reproducing kernel are the four conditions in Theorem 5.13 equivalent? What about just the three conditions 1, 3 and 4 (it is widely suspected that these three conditions are always equivalent)? 5.2 Higher dimensions We now consider the characterization of interpolating sequences for the Hilbert spaces B2" (lB n ) , 0 ::; (1 < ~, as well as the corona theorem for the corresponding multiplier algebras MB~(Bn)' As the techniques used are largely based on the theory of Hilbert spaces, we will also investigate two other problems related to Hilbert space theory; namely von Neumann's inequality and the characterization of multiplier algebras. These problems all involve the notion of Carleson measures, to which we

10.i

5.2. Higher dbneIlJllo....

now turn. The material on Carleson measures presented here is taken largely from [4], [5J and [6].

5.2.1 Carleson Measures. By a Carleson measure for B 2(18 n ) we mean a positive measure defined on 18n such that the following Carleson embedding holds for f E B 2(18 n ),

fen If(z)1 2 dp, ~ CI-' Iifli~~(Bn) .

(5.22)

The set of all such is denoted CM(B2(18 n » and we define the Carleson measure norm 1iP,ll carleson to be the infimum of the possible values of C;/2. In [5] we described the Carleson measures for B2 (18 n ) for a = O. Here we consider a ~ O. We show that, mutatis mutandis, the results for a = 0 extend to the range 0 ~ a < 1/2. FUndamental to this extension is the fact that for o ~ a < 1/2 the real part of the reproducing kernel for B 2(18 n ) is comparable to its modulus. However, even though the reproducing kernel for B;/2(18 n ) has positive real part, its real part is not comparable to its modulus. For that space a new type of analysis must be added - see [6]. For 1/2 < a < n/2 the real part of the kernel is not positive, our methods don't apply, and that range remains mysterious. For a ~ n/2 we are in the realm of the classical Hardy and Bergman spaces and the description of the Carleson measures is well established [39], [53J. We will need to extend the dyadic tree T in the disk to the ball in higher dimensions as in [5].

5.2.2 Construction of the Bergman tree. Let (3 be the Bergman metric on the unit ball:IB n in en. Note that the set

Sr

= oB{3(O,r) = {z

E 18n

:

(3(O,z)

=

r}

is a Euclidean sphere (with different radius) centered at the origin for each r In fact, by (lAO) in [53] we have (3(0, z) = tanh-Ilzl, and so

1 - Izl2 = 1 - tanh2 (3(O, z) 4 - e 2 {J(o.z) + 2 + e- 2 {3(O,z) ~

> O.

(5.23)

4e- 2{3(o.z)

for (3(O, z) large. We recall the following elementary abstract construction from [5] (Lemma 7 on page 18)

Lemma 5.19 Let (X, d) be a separable metric space and)" > O. There is a denumerable set of points E = {xJ }~r J and a corresponding set of Borel subsets Qj of X satisfying X --

OO or JQ u j=l j,

Qi n Qj =
c QJ c

i

(5.24)

=I j,

B(Xj, 2),,),

j ~ l.

We refer to the sets QJ as unit qubes centered at Xj' In [5], we applied Lemma 5.19 to the spheres Sr for r > 0 as follows. Fix 0,).. > 0 (vaxious choices of 0 and ).. will be used below), which we will refer to as structural constants for the Bergman tree. For N E N, apply the lemma to the metric space (SN9, {3} to obtain points {ZJ"}f=l and unit ~ub~s {Qf};=l in SN9 satisfying (5.24). For z E :IBn, let Prz denote the radial proJectIon of z onto the sphere Sr (not to be confused with

li

104

Hilbert Function Spaces and Nevanlinna-Pick Kernels

Pa defined above). We now define subsets Kj' K? = {z E lffin . /3(0, z) < O} and Kf = {z E lffin : NO ::; d(O, z) < (N + 1)8, PN(Jz E Qf}, N ~ I,J We define corresponding points c~ E K j' by the orthogonal projection

cf =

of lffin by ~ 1

p(N+!)o(zf)

as

We will refer to the subset Kj' of lffin a unit kube centered at centered at 0) Define a tree structure on the colledion of unit kubes

Tn = {K~}N~O 1~1 by declaring that Kfl+l is a child of Kf, written Kf"+1

cJ'

(while

K?

is

C(Kn, if the projection PNO(zf"+l) of zf"+l onto the sphere SNO lies in the qube In the case N = 0, we declare every kube KJ to be a child of the root kube K? We will typically write a. /3, 'Y etc. to denote elements Kf of the tree Tn when the corre5pondence with the unit ball lffin is immaterial. We will write K<> for the kube K: and Co: for its center when the correhpondence matters. We thus have t3 > a if a = K;:' N and /3 = K J+'M> and there is a sequence of kubes {K,!H}M f>uch that KNH is 4' )1 £=1' 1£ a child of K;:~/:-1 fO! 1 ~ " ~ M Sometimes we will further abuse notation by using a to denote the eenter Ca = of the kube K" = Kj"', especially in the section on interpolating :,equences below Finally, we recall the dimension neT) of an arbitrary tree T. E

Qf

cf

cf

Definition 5.20 The upper dimenswn neT) of a tree T if-> given by neT) = lim sup log2(Np)J, £---'>00

where Ne

= 5Up card{fJ E T· /3 > a and

d({3) = dCd')

+ f},

nET

along with a similar definition for the lower dimension '!leT) using lim infr_oc in place of lim sUPe_oo If the upper and lower dimensions coincide, we denote their common value, called the dimenston of T, by neT) Note that if T i:, a homogeneous tree with branching number N, then N = (Ne) ~ for all £ :::=: 1. The choice of base 2 for the logarithm then yields the r elatiOllship N = 2n , consistent with the familiar interpretation that the dyadic tree ha:, dimension 1 and the linear tree has dimension The proof of the following lemma is in [5J

°

Lemma 5.21 The tree Tn, con~tructed above with positive parameters A and 0, and the unit balllffin satisfy the followmg properties 1 The balllffi" is a pairwise diSjoint union of the kubes Ko:, a E 7". and there are posztive constants C 1 and C 2 dependmg on A and 0 such that

B{3(co" C 1 ) C Ko. c B{3(co., C 2 ),

a E

2 U{3~o. K(3 is "comparable" to the Carles on tent co., where Vz = {w E lffin . /1 - 'ill. Pzl ::; 1 -

Tn, n :::=: 1. v;." associated Izl},

to the point

105

5.2. Higher dimensions

and pz denotes radial projection of z onto the sphere 8lffi n . More precisely "comparable" means there are positive constants aI, a2 such that al Vee. C U,6~a K{:J C a2Vca where rVz == Vz(r) and z(r) is the positive multiple of z satisfying I-iz(r)i = r I-izi

3 The invariant volume of Ka is bounded between positive constants depending on ).. and 0, but independent of 0: E Tn. 4 The dimension n(7;,) of the tree 7;, is 1~82 n S For any R > 0, the balls B(:J(c n . R) satisfy the finite overlap condition

L

ABl(r" R)(Z) S;

CR ,

z E lffi n .

"ETn

5.2.3 Characterization of Carleson measures on the ball. Now we return briefly to an investigation of Carleson measures for B2(lffin)' As the proofs are rather lengthy, we postpone them to the following section, and content ourselves here to careful statements of the results for use in the applications deo,cribed in this section. Let Tn denote the Bergman tree constructed above. We show in the next section (Theorem 6 9 (p. 141)) that the tree condition,

L

[20"d(6) 1* p(8)]2 S;

C 1* p( ex) <

00,

exETn,

(S 2S)

32':<>

characterizes Carleo,on meao,ures for the Besov-Sobolev space B 2 (lffi n ) in the range o S; u < 1/2 Here I*p(ex) = "E.(:I>cx fJ (f3) On the other hand, if u ;::: 1/2, then by the results in [IS], there io, a positive measure p on the ball that is Carleson for J~_<7(lffin) = B2'(lffi,,), but that fails to o,atisfy the tree condition (5 2S) 2

Our analysis of Carleson measures for the "endpoint" cao,e B~/\lffin)' the DruryArveo,on Hardy space, proceeds in two stages. First, by a function analytic argument we show that the norm iifJllcarlesoTl is comparable, independently of dimension, with the best COU'3tant C in the inequality

ll" In

(Re 1 _

~. Z' )f(ZI)dII(Z')g(z)dp(z) \ S; C Ilfllp(fL) Ilgllp(fL)

(5.26)

We then proceed with a geometric analysis of the conditions under which this inequality holdo, If in (S 26) we were working with the integration kernel

I-i

II-i I Zl

lather than Re Zl we could do an analysis similar to that for u < 1/2 and would hnd the inequality characterized by the tree condition with u = 1/2 .

L[2 d (p)/21*fJ(fJW

S;

C1*p(ex) <

00.

(5.27)

p~a

However, as we will show, the finiteness of Ilpllcarle,on is equivalent neither to the tree condition (527), nor to the simple condition ex E 7;,.

(5.28)

To proceed we mUbt introduce additional structure on the Bergman tree Tn For ex in Tn, we denote by [ex] an equivalence class in a certain quotient tree Rn of "rings" consisting of elements in a "common slice" of the ball having common distance flOm the root. Using this additional structure it is shown in [6] that the Carleson

106

5. Hilbert Function Spaces and Nevanlinna-Pick Kernels

measures are characterized by the simple condition (5.28) together with the "split" tree condition

LL

L

2d (-y)-k

k;?:O ";?:,,,

I*J.L(8)I*J.L(8') ~ CI*J.L(a),

a

E

'Tn,

(5.29)

(o,o')EQ(k) h)

and moreover we have the norm estimate, (5.30)

+

sup ",ET" r*/-I(a»O

The restriction (8,8') E g(k)(-y) in the sums above means that we sum over all pairs (8,8') of grandk-children of '"'( that have '"'( as their minimum, that lie in separate rings in the quotient tree, but whose predecessors of order two lie in a common ring. Note that if we were to extend the summation to all pairs (8, 8') of grand k -children of "'f then this condition would be equivalent to the tree condition (5.27). More formally, if a /\ 13 denotes the largest tree element that is equal to or less than both a and 13, Definition 5.22 The set g(k)(-y) consists of pailS (8,8') of grandk-children of'"'( in g(k)(,",() x g(k)(-y) which satisfy 8/\8' = '"'(, [A 28] = [A 28'] (which implies d([8], [8']) ~ 4) and d* ([8], [8']) = 4. Here

d*([a], [13])

=

inf d([t(Uca )], [t(UcJ3)]),

UEU"

and Un denotes the group of unitary rotations of the ball lmn For a in the Bergman tree Tn, Ca is the "center" of the Bergman kube K", For Z E lm n , t(z) denotes the element a' E 'Tn such that z E K a ,. Thus d*([a], [13]) measures an "invariant" distance between the rings [a] and [13]. Note that g(O)(-y) = g(-y) is the set of grandchildren of '"'(. The condition (5.29) can be somewhat simplified by further restricting the sum over k and '"'( on the left side to k ~ cd(-y) for any fixed c > 0, the resulting c-split tree condition is

The reason (5.28) and (5.31) suffice is that the sum in (5.29) over k > cd(-y) is dominated by the left side of (5.25) with a = (1-c)/2, and that this condition is in turn implied by the simple condition (5.28) - see [6]. Here is a precise statement of the characterization of Carleson mearures for the Drury-Arveson Hardy space H~. Theorem 5.23 ([6]) A positive measure J.L on the ball lmn is H~-Carleson if and only if J.L satisfies the simple condition (5.28) and the split tree condition (5.29) taken over all unitary rotations of a fixed Bergman tree. Moreover, the estimate (5.30) holds where the supremum is also taken over all unitary rotations Tn of a fixed Bergman tree.

5.3. Applications of Carl......n

Dl......U ......

Remark 5.24 We can recast the above characterization on the ball as follows. The H;-Carleson norm of a positive measure /L on $nsatisfies Cn

1I/Ll!carleson::; sup WEB"

+

J(l -lwI )-11t(T(w)) 2

(T~ w» /L

sup wEBn p(T(w»>O

I

f f (Re 1 JT(w} JT(w)

1

z .z

2

l)dlt (ZI)1 d/L(z)

::; C n IIltliCarleson .

In this formulation, Theorem 5.23 has been recently obtained by Tchoundja (48J using techniques inspired partly by the effective use of Menger curvature in studying fractional bingular integrals in single variable function theory. Finally, as we mentioned, the characterization of Carleson measures for B2 (Bn) remains open in the range 1/2 < u < n/2. The Carleson measures for the Hardy space, u = n/2, and the weighted Bergman spaces, u > n/2, are characterized by the analogue of the simple condition (5.28) [39J, [53J. 5.3 Applications of Carleson measures

It is well known that Carleson measures often arise as part of the answer to natural questions about function spaces such as B 2{B n ). In this subsection we give two iru.tances of that First we describe the multiplier algebra MB~(Bn) of B2($n) for 0 (1 1/2. Second we give an explicit formula for the norm which arises in Drury's generalization of von Neumann's operator inequality to the complex ball Bn In the next subsection we describe the interpolating sequences for B2 (Bn) and for M B~ (B .. ) for the range 0 u < 1/2 530.1 Multipliers. A holomorphic function 1 on the ball is called a multiplier fox the bPace B~($,,) if the multiplication operator Mf defined by Mf(9) = Ig is a bounded linear operator on B~($n) - see Definition 5.27 below. In that case the multiplier norm of 1 is defined to be the operator norm of Mf(9) The space of all such is denoted MB~(I8,,) With the natural multiplication MBr(s,,) is a commutative Banach algebra Ortega and Fahrega [33] have characterized multipliers for the Hardy-Sobolev spaces using Carlebon measureb \Ve refine their result by including a geometric characterization of those measures

:s :s

:s

:s :s

Theorem 5.25 Suppose 0 u 1/2 Then 1 is in is bounded and for some, eqUivalently for any, m > n/2 d/Lj,k

= !(l_l z I2 )ffl+U f(m)(z)j2 dAn(Z)

E

Mml(Bn)

if and only if f

(1

CM(B2"(B n ).

In that case we have

I!fIlMB~(lIIfl) ~ IIfIl H oo(s,,) + Ildltf,kllcM(BHBn»

:s

If 0 u < 1/2 the second summand can be evaluated using Theorem 6.9. For u = 1/2 the second summand can be evaluated using Theorem 5.23 Remark 5.26 In the most familiar case, the one variable Hardy space, n = 1, = 1; the Carleson mea.&ure condition is usually not mentioned

u = 1/2, and m

108

5. Hilbert Function Spaces and Nevanlinna-Pick Kernels

That is because in that case the Carleson measure condition is implied by the boundedness of I, for instance because of the inclusion HOO(Rt} C BMO(R 1 )

Definition 5.21 We say that