What Hilbert spaces can tell us about bounded functions on the bidisk

0.2 MB·English

by  Jim Agler

What Hilbert spaces can tell us about bounded
9 functions in the bidisk
0
0
2 Jim Agler and John E. McCarthy

n
a
Dedicated to the memory of Paul Halmos
J

7
Abstract. We discuss various theorems about bounded analytic functions on
] thebidisk that were proved using operator theory.
V
C
.
h
t
a
m 1. Introduction
[

Much of modern single operator theory (as opposed to the study of operator al-
1
v
gebras) rests on a foundation of complex analysis. Every cyclic operator can be
7
0 representedasmultiplicationbytheindependentvariableonthecompletionofthe
9
0 polynomialswithrespecttosomenorm.ThenicestwaytohaveanormisinL2(µ),
.
1 andthen one is ledto the study ofsubnormaloperators,introducedby P.Halmos
0
9 in [34]. The study of cyclic subnormal operators becomes the study of the spaces
0
: P2(µ), the closure of the polynomials in L2(µ), and the theory of these spaces re-
v
i lies on a blend of complex analysis and functional analysis; see J. Conway’s book
X
r [27] for an exposition. Alternatively, one can start with a Hilbert space that is
a
amenable to complex analysis, such as the Hardy space H2, and study classes of
operators on that space that have a good function theoretic representation, such
asToeplitz,Hankelorcompositionoperators.Alloftheseclassesofoperatorshave
Jim Agler was partially supported by National Science Foundation Grant DMS 0400826, and
JohnMcCarthywaspartiallysupportedbyNationalScienceFoundation GrantDMS0501079.
2 Jim Agler and John E. McCarthy
arichtheory,whichdependsheavilyonfunctiontheory—forexpositions,seee.g.
[41, 43] and [29].
The traffic, of course,goes both ways. There are many questions in function
theory that have either been answered or illuminated by an operator theory ap-
proach. The purpose of this article is to describe how operator theory has fared
when studying H∞(D2), the algebra of bounded analytic functions on the bidisk
D2.Wefocusonfunctiontheoryresultsthatwereproved,originallyatleast,using
operator theory.
For the topics in Sections 2 to 7, we shall first describe the situation on the
diskD,andthenmoveontothe bidisk.The topicsinSections8to9donotreally
have analogues in one dimension. For simplicity, we shall stick to scalar-valued
functiontheory,thoughmanyoftheresultshavenaturalmatrix-valuedanalogues.
We shall use the notation that points in the bidisk are called λ or ζ, and
their coordinateswill be givenby superscripts:λ=(λ1,λ2). We shalluse z andw
to denote the coordinate functions on D2. The closed unit ball of H∞(D) will be
written H∞(D) and the closed unit ball of H∞(D2) as H∞(D2).
1 1
2. Realization Formula
The realization formula is a way of associating isometries (or contractions) with
functions in the ball of H∞(D). In one dimension, it looks like the following; see
e.g. [19] or [9] for a proof.
Theorem 2.1. The function φ is in the closed unit ball of H∞(D) if and only if
there is a Hilbert space and an isometry V :C C , such that, writing
H ⊕H→ ⊕H
V as
C
H
C A B
V = , (2.2)
 
C D
H
one has  
φ(λ) = A+λB(I λD)−1C. (2.3)
−
What Hilbert spaces can tell us about bounded functions in the bidisk 3
This formula was generalized to the bidisk in [3]. It becomes
Theorem 2.4. The function φ is in the closed unit ball of H∞(D2) if and only if
there are auxiliary Hilbert spaces and and an isometry
1 2
H H
V : C C
1 2 1 2
⊕H ⊕H → ⊕H ⊕H
such that, if := , V is written as
1 2
H H ⊕H
C
H
C A B
V = , (2.5)
 
C D
H
and =λ1I λ2I , then  
Eλ H1 ⊕ H2
φ(λ) = A+B (I D )−1C. (2.6)
λ H λ
E − E
There is a natural generalization of (2.6) to functions of d variables. One
chooses d Hilbert spaces ,..., , lets = , lets = λ1I
H1 Hd H H1⊕···⊕Hd Eλ H1 ⊕
λdI , and then, for any isometry V as in (2.5), let
···⊕ Hd
ψ(λ) = A+B (I D )−1C. (2.7)
λ H λ
E − E
Thesetofallfunctionsψ thatarerealizableinthiswayisexactlythe Schur-Agler
class, a class of analytic functions of d variables that can also be defined as
ψ : ψ(T ,...,T ) 1 commuting contractive matrices (T ,...,T ) .
1 d 1 d
{ k k≤ ∀ }
(2.8)
VonNeumann’sinequality[58]istheassertionthatford=1theSchur-Aglerclass
equals H∞(D); Andˆo’s inequality [16] is the equality for d = 2. Once d > 2, the
1
Schur-Agler class is a proper subset of the closed unit ball of H∞(Dd) [57, 30].
Many of the results in Sections 3 to 7 are true, with similar proofs, for the Schur-
Agler class in higher dimensions (or rather the norm for which this is the unit
ball)1, but it is not know how to generalize them to H∞(Dd).
The usefulness of the realizationformulastems primarily notfromits ability
to represent functions, but to produce functions with desired properties with the
1Specifically,Theorems3.7,4.3,6.4and(i)⇔(iii)ofTheorem7.7.
4 Jim Agler and John E. McCarthy
aid of a suitably chosen isometry V (dubbed a lurking isometry by Joe Ball). An
exampleofalurkingisometryargumentistheproofofPick’stheoreminSection3.
It is well-known that equality occurs in the Schwarz lemma on the disk only
for Mo¨bius transformations. The Schwarz lemma on Dd is the following (see [47]
for a proof).
Proposition 2.9. If f is in H∞(Dd), then
1
d
∂f
(1 λr 2) (λ) 1 f(λ)2. (2.10)
−| | ∂λr ≤ −| |
r=1 (cid:12) (cid:12)
X (cid:12) (cid:12)
(cid:12) (cid:12)
G. Knese [38]provedthat equ(cid:12)ality in ((cid:12)2.10)resultedin a curious formofthe
realizationformula.(Noticethatford 3,thehypothesisisthatf lieinH∞(Dd),
≥ 1
but the conclusion means f must be in the Schur-Agler class.)
Theorem2.11. Suppose f H∞(Dd) is afunction of all dvariables, and equality
∈ 1
holds in (2.10)everywhere on Dd. This occurs if and only if f has a representation
asin(2.7)whereeachspace isone-dimensional,andtheunitaryV issymmetric
r
H
(equal to its own transpose).
Analyzingtherealizationformula,J.M.Anderson,M.DritschelandJ.Rovnyak
were able to obtain the following higher derivative version of the Schwarz lemma
[15]:
Theorem 2.12. Let f be in H∞(D2), and n ,n non-negative integers with n =
1 1 2
n +n . Let λ=(z,w) be in D2, with λ =max(z , w). Then
1 2
| | | | | |
∂nf 1 f(λ)2 n2 n 2n n n2 n
(n 2)! −| | 1− 1 + 1 2 + 2− 2 .
(cid:12)∂n1z∂n2w (cid:12) ≤ − (1−|λ|)n−1 "1−|z|2 1−|z|2 1−|w|2 1−|w|2#
(cid:12) (cid:12)
(cid:12) (cid:12) p p
(cid:12) (cid:12)
3. Pick Problem
ThePickproblemonthe diskistodetermine,givenN points λ ,...,λ inDand
1 N
N complex numbers w ,...,w , whether there exists φ H∞(D) such that
1 N ∈ 1
φ(λ )=w , i=1,...,N.
i i
What Hilbert spaces can tell us about bounded functions in the bidisk 5
G. Pick proved [44] that the answer is yes if and only if the N-by-N matrix
1 w w¯
i j
− (3.1)
1 λ λ¯
(cid:18) − i j (cid:19)
is positive semi-definite.
D. Sarason realized in [49] that Pick’s theorem could be proved by showing
that operators that commuted with the backward shift on an invariant subspace
could be lifted to operators that commute with it on all of H2; this result was
then generalized by B. Sz-Nagy and C. Foia¸s to the commutant lifting theorem
[52]. Here is a proof of Pick’s theorem using a lurking isometry.
Proof: If (3.1) is of rank M, then one can find vectors g N in CM such
{ i}i=1
that
1 w w¯
i j
1− λ λ¯ = hgi,gjiCM. (3.2)
i j
−
We can rewrite (3.2) as
1+ λ g ,λ g = w w¯ + g ,g . (3.3)
i i j j i j i j
h i h i
The lurking isometry V :C CM C CM is defined by
⊕ → ⊕
1 w
V : i . (3.4)
λigi 7→ gi
(cid:18) (cid:19) (cid:18) (cid:19)
We extend linearly to the span of
1
: i=1,...,N , (3.5)
λ g
i i
(cid:26)(cid:18) (cid:19) (cid:27)
and if this is not the whole space C CM, we extend V arbitrarily so that it
⊕
remains isometric. Write V as
C CM
C A B
V = ,
CM C D 
 
and define φ by
φ(λ) = A+λB(I λD)−1C. (3.6)
−
6 Jim Agler and John E. McCarthy
By the realization formula Thm. 2.1, φ is in H∞(D). Moreover, as (3.4) implies
1
that
A+Bλ g = w
i i i
C+Dλ g = g ,
i i i
we get that
(I λ D)−1C = g ,
i i
−
and hence
φ(λ ) = A+λ Bg = w ,
i i i i
so φ interpolates. 2
(It is not hard to show that φ is actually a Blaschke product of degree M).
A similar argument using Thm. 2.4 solves the Pick problem on the bidisk.
The theorem was first proved, by a different method, in [2].
Theorem 3.7. Given points λ ,...,λ in D2 and complex numbers w ,...,w ,
1 N 1 N
there is a function φ H∞(D2) that maps each λ to the corresponding w if and
∈ 1 i i
only if there are positive semi-definite matrices Γ1 and Γ2 such that
1 w w¯ = (1 λ1λ¯1)Γ1 +(1 λ2λ¯2)Γ2 . (3.8)
− i j − i j ij − i j ij
On the polydisk, a necessary condition to solve the Pick problem analagous
to (3.8) has recently been found by A. Grinshpan, D. Kaliuzhnyi-Verbovetskyi,
V. Vinnikov and H. Woerdeman [33]. As of this writing, it is unknown if the
condition is also sufficient, but we would conjecture that it is not.
Theorem3.9. Given points λ ,...,λ in Dd and complex numbers w ,...,w , a
1 N 1 N
necessary condition for there to be a function φ H∞(Dd) that maps each λ to
∈ 1 i
the corresponding w is: For every 1 p < q d, there are positive semi-definite
i
≤ ≤
matrices Γp and Γq such that
1 w w¯ = (1 λrλ¯r)Γq + (1 λrλ¯r)Γp. (3.10)
− i j − i j ij − i j ij
r6=q r6=p
Y Y
What Hilbert spaces can tell us about bounded functions in the bidisk 7
4. Nevanlinna Problem
If the Pick matrix (3.1) is singular (i.e. if M < N) then the solution is unique;
otherwise it is not. R. Nevanlinna found a parametrization of all solutions in this
latter case [40] (see also [20] for a more modern approach).
Theorem 4.1. If (3.1) is invertible, there is a 2-by-2 contractive matrix-valued
function
G G
G = 11 12
G G
21 22
(cid:18) (cid:19)
such that the set of all solutions of the Pick problem is given by
ψG
φ=G +G 21 : ψ H∞(D) .
{ 11 121 G ψ ∈ 1 }
22
−
On the bidisk, we shall discuss uniqueness in Section 8 below. Consider now
the non-unique case. Let φ be in H∞(D2), and so by Theorem 2.4 it has a repre-
1
sentation as in (2.6). Define vector-valued functions F ,F by
1 2
C
F (λ)
H1 1 := (IH D λ)−1C. (4.2)
  − E
F (λ)
2 2
H
 
For a given solvable Pick problem with a representation as (3.8), say that φ is
affiliated with (Γ1,Γ2) if, for some representation of φ and F ,F as in (4.2),
1 2
F (λ )∗F (λ ) = Γ1
1 i 1 j ij
F (λ )∗F (λ ) = Γ2
2 i 2 j ij
for i,j =1,...N. The situation is complicated by the fact that for a given φ, the
pairs (Γ1,Γ2) with which it is affiliatedmay or may not be unique. J. Ball and T.
Trent [21] proved:
Theorem 4.3. Given a solvable Pick problem, with a representation as in (3.8),
there is a matrix-valued function G
C CM
C G G
G = 11 12
CM G G 
21 22
 
8 Jim Agler and John E. McCarthy
in the closed unit ball of H∞(D2,B(C CM,C CM)), such that the function
⊕ ⊕
φ solves the Pick problem and is affiliated with (Γ1,Γ2) if and only if it can be
written as
φ = G +G Ψ(I G Ψ)−1G (4.4)
11 12 22 21
−
for some Ψ in H∞(D2,B(CM,CM)).
1
5. Takagi Problem
The case where the Pick matrix (3.1) has some negative eigenvalues was first
studiedbyT.Takagi[54],andlaterbymanyotherauthors[1,42,20].Seethebook
[19]foranaccount.Theprincipaldifferenceisthatifonewishestointerpolatewith
a unimodular function(i.e. afunction that hasmodulus oneonthe unit circleT),
then one has to allow poles inside D. A typical result is
Theorem5.1. Supposethe Pick matrix is invertible, andhas π positive eigenvalues
andν negativeeigenvalues.Thenthereexistsameromorphicinterpolatingfunction
φ that is unimodular, and is the quotient of a Blaschke product of degree π by a
Blaschke product of degree ν.
If Γ is not invertible, the problem is degenerate. It turns out that there is a
big difference between solving the problem of finding Blaschke products f,g such
that
f(λ ) = w g(λ )
i i i
and the problem of solving
f(λ )/g(λ ) = w .
i i i
(Thedifferenceoccursiff andg bothvanishatsomenodeλ ;inthe firstproblem
i
theinterpolationconditionbecomesvacuous,butinthesecondoneneedsarelation
on the derivatives). The first problem is more easily handled as the limit of non-
degenerateproblems;see the paper [23]for recentdevelopments onthis approach.
Thesecondversionoftheproblemhasonlyrecentlybeeensolved,byH.Woracek,
using Pontryjagin spaces [59].
What Hilbert spaces can tell us about bounded functions in the bidisk 9
Question 1. What is the right version of Theorem 5.1 on the bidisk?
6. Interpolating Sequences
Givenasequence λ ∞ inthepolydiskDd,wesayitisinterpolatingforH∞(Dd)
{ i}i=1
if, for any bounded sequence w ∞ , there is a function φ in H∞(Dd) satisfying
{ i}i=1
φ(λ )=w . L. Carleson characterized interpolating sequences on D in [24].
i i
Beforestating his theorem,let us introduce some definitions.A kernel onDd
is a positive semi-definite function k :Dd Dd C, i.e. a function such that for
× →
any choice of λ ,...,λ in Dd and any complex numbers a ,...,a , we have
1 N 1 N
a a¯ k(λ ,λ ) 0.
i j i j
≥
X
Given any kernel k on Dd, a sequence λ ∞ has an associated Grammian Gk,
{ i}i=1
where
k(λ ,λ )
[Gk] = i j .
ij
k(λ ,λ )k(λ ,λ )
i i j j
We think of Gk as an infinite matrix, representing an operator on ℓ2 (that is not
p
necessarily bounded). When k is the Szego˝ kernel on Dd
1
kS(ζ,λ) = , (6.1)
(1 ζ1λ¯1)(1 ζ2λ¯2) (1 ζdλ¯d)
− − ··· −
wecalltheassociatedGrammiantheSzego˝ Grammian.TheSzego˝kernelisthere-
producingkernelforthe HardyspaceH2(Dd)=P2(m),wherem isd-dimensional
Lebesgue measure on the distinguished boundary Td of Dd.
An analogue of the pseudo-hyperbolic metric on the polydisk is the Gleason
distance, defined by
ρ(ζ,λ) := sup{|φ(ζ)|:kφkH∞(Dd) ≤1,φ(λ)=0}.
We shall call a sequence λ ∞ weakly separated if there exists ε > 0 such that,
{ i}i=1
for all i = j, the Gleason distance ρ(λ ,λ ) ε. We call the sequence strongly
i j
6 ≥
separated if there exists ε>0 such that, for all i, there is a function φ in H∞(D)
i 1
such that
ε, j =i
φ (λ ) =
i j 0, j =i
(cid:26) 6
10 Jim Agler and John E. McCarthy
In D, a straightforwardargument using Blaschke products shows that a sequence
is strongly separated if and only if
ρ(λ ,λ ) ε i.
i j
≥ ∀
j6=i
Y
We can now state Carleson’s theorem. Let us note that he proved it us-
ing function theoretic methods, but later H. Shapiro and A. Shields [51] found a
Hilbert space approach, which has proved to be more easily generalized, e.g. to
characterizing interpolating sequences in the multiplier algebra of the Dirichlet
space [39].
Theorem 6.2. On the unit disk, the following are equivalent:
(1) There exists ε>0 such that
ρ(λ ,λ ) ε i.
i j
≥ ∀
j6=i
Y
(2) The sequence λ ∞ is an interpolating sequence for H∞(D).
{ i}i=1
(3) The sequence λ ∞ is weakly separated and the associated Szego˝ Gram-
{ i}i=1
mian is a bounded operator on ℓ2.
In 1987 B. Berndtsson, S.-Y. Chang and K.-C. Lin provedthe following the-
orem [22]:
Theorem 6.3. Let d 2. Consider the three statements
≥
(1) There exists ε>0 such that
ρ(λ ,λ ) ε i.
i j
≥ ∀
j6=i
Y
(2) The sequence λ ∞ is an interpolating sequence for H∞(Dd).
{ i}i=1
(3) The sequence λ ∞ is weakly separated and the associated Szego˝ Gram-
{ i}i=1
mian is a bounded operator on ℓ2.
Then (1) implies (2) and (2) implies (3). Moreover the converse of both these
implications is false.