The Hardy space of a slit domain

The Hardy space of a slit domain A l e x a n d r u Aleman, Nathan S. Feldman, William T. Ross Preface If H is a Hilbe...

Author: Alexandru Aleman | Nathan S. Feldman | William T. Ross

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The Hardy space of a slit domain A l e x a n d r u Aleman, Nathan S. Feldman, William T. Ross

Preface If H is a Hilbert space and T : H → H is a continous linear operator, a natural question to ask is: What are the closed subspaces M of H for which T M ⊂ M? Of course the famous invariant subspace problem asks whether or not T has any non-trivial invariant subspaces. This monograph is part of a long line of study of the invariant subspaces of the operator T = Mz (multiplication by the independent variable z, i.e., Mz f = z f ) on a Hilbert space of analytic functions on a bounded domain G in C. The characterization of these Mz -invariant subspaces is particularly interesting since it entails both the properties of the functions inside the domain G, their zero sets for example, as well as the behavior of the functions near the boundary of G. The operator Mz is not only interesting in its own right but often serves as a model operator for certain classes of linear operators. By this we mean that given an operator T on H with certain properties (certain subnormal operators or two-isometric operators with the right spectral properties, etc.), there is a Hilbert space of analytic functions on a domain G for which T is unitarity equivalent to Mz . Probably the first to successfully study these types of problems was Beurling [13] who gave a complete characterization of the Mz -invariant subspaces of the Hardy space n of the unit disk. These are the functions f (z) = ∑∞ n=0 an z which are analytic on the open 2 unit disk D := {|z| < 1} for which ∑n>0 |an | < ∞. Many others followed with a discussion, often a complete characterization, of the Mz -invariant subspaces where the Hardy n space is replaced by the space of analytic functions f (z) = ∑∞ n=0 an z on D satisfying 2 ∑n>0 wn |an | < ∞, where (wn )n>0 is a sequence of positive weights. For example, when wn = n, we get the classical Dirichlet space where the Mz -invariant subspaces were discussed in [61, 62, 63]. When wn = nα and α > 1, we get certain weighted Dirichlet spaces where the Mz -invariant subspaces were completely characterized in [70]. See [53, 54] for some related results. When wn = n−1 (or more generally wn = nα , α < 0), we get the Bergman (weighted Bergman) spaces where the Mz -invariant subspaces were discussed in [8, 69]. See also [31, 43]. In Beurling’s seminal paper, and the ones that followed, notice how the underlying domain of analyticity is kept fixed to be the unit disk D, but the Hilbert space of analytic functions is changed by varying the weights wn . In a series of papers beginning with Sarason [66], the basic type of Hilbert space is fixed but the domain of analyticity is changed. To see what we mean here, the condition f (z) = ∑n>0 an zn is analytic on D and ∑n>0 |an |2 < ∞, the definition of the Hardy space of D, can be equivalently restated as

vi

Preface

f is analytic on D and there is a harmonic function U on D for which | f |2 6 U on D. Such a function U is called a harmonic majorant for | f |2 . For a general bounded domain G ⊂ C, one can define the Hardy space of G to be the analytic functions f on G for which | f |2 has a harmonic majorant on G. Beginning with Sarason’s paper, there were several authors [6, 7, 38, 45, 65, 77, 78, 79] who characterized the Mz -invariant subspaces of the Hardy space of annular-type domains, which include an annulus, a disk with several holes removed, and a crescent domain (the region between two internally tangent circles). Conspicuously missing from this list of domains are slit domains, for example G = D \ [0, 1). In this monograph, we obtain a complete characterization of the Mz invariant subspaces of the Hardy space of slit domains. Along the way, we give a thorough exposition of the Hardy space, and even the Hardy-Smirnov space, of a slit domain as well as several applications of our results to de Branges-type spaces and the classical backward shift operator of the Hardy space of D. We also discuss several aspects of the operator Mz |M, where M is an Mz -invariant subspace of the Hardy space of G. In particular, we explore questions about cyclicity, the spectrum, and the essential spectrum for Mz |M.

Contents Preface

v

Notation

ix

List of Symbols

xi

Preamble

xiii

1 Introduction 1.1 Some history . . . . . . . . . . . 1.2 Invariant subspaces of the slit disk 1.3 Nearly invariant subspaces . . . . 1.4 Cyclic invariant subspaces . . . . 1.5 Essential spectrum . . . . . . . .

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1 1 2 5 6 7

2 Preliminaries 2.1 Hardy space of a general domain 2.2 Harmonic measure . . . . . . . 2.3 Slit domains . . . . . . . . . . . 2.4 More about the Hardy space . .

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9 9 12 14 20

3 Nearly invariant subspaces 3.1 Statement of the main result . . 3.2 Normalized reproducing kernels 3.3 The operator J . . . . . . . . . . 3.4 The Wold decomposition . . . . 3.5 Proof of the main theorem . . . 3.6 Uniqueness of the parameters . .

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25 25 26 34 37 42 46

4 Nearly invariant and the backward shift 4.1 The backward shift and pseudocontinuations . . . . . . . . . . . . . . . . 4.2 A new description of nearly invariant subspaces . . . . . . . . . . . . . .

47 47 48

Contents

viii

5 Nearly invariant and de Branges spaces 5.1 de Branges spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 de Branges spaces and nearly invariant subspaces . . . . . . . . . . . . .

59 59 60

6 Invariant subspaces of the slit disk 6.1 First description of the invariant subspaces . . . . . . . . . . . . . . . . . 6.2 Second description of the invariant subspaces . . . . . . . . . . . . . . .

65 65 68

7 Cyclic invariant subspaces 7.1 Two-cyclic subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Cyclic subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Polynomial approximation . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 80 82

8 The essential spectrum 8.1 Fredholm theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Essential spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 85

9 Other applications 9.1 Compressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 93 95

10 Domains with several slits 10.1 Statement of the result . . . 10.2 Some technical lemmas . . . 10.3 A localization of Yakubovich 10.4 Finally the proof . . . . . .

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97 . 97 . 99 . 101 . 111

11 Final thoughts

113

12 Appendix

115

Notation The complete list of symbols is contained in the next chapter. Below are some basic symbols and remarks regarding notation and organization. • C (complex numbers)

b = C ∪ {∞} (Riemann sphere) • C • R (real numbers)

• D = {z ∈ C : |z| < 1}

• T = ∂ D = {z ∈ C : |z| = 1} • N = {1, 2, · · ·} • N0 = {0, 1, 2, · · ·} • When defining functions, sets, operators, etc., we will often use the notation A := xxx. By this we mean A ‘is defined to be’ xxx. • As is traditional in analysis, the constants c, c′ , c′′ , · · · c1 , c2 , · · · can change from one line to the next without being relabeled. • Numbering is done by chapter and section, and all equations, theorems, propositions, and such are numbered consecutively. • If J is a set in some topological vector space, elements of J and J − is the closure of J.

W

J is the closed linear span of the

• If A ⊂ C, then A = {a : a ∈ C}, the complex conjugate of the elements of A. From the previous item, note that A− is the closure of A. • A linear manifold in some topological vector space is a set which is closed under the basic vector space operations. A subspace is a closed (topologically) linear manifold.

List of Symbols S (shift operator S f = z f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 1 D (open unit disk) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 1 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 2 T (unit circle) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 3 H 2 (Ω) (Hardy space of a domain Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 9 U f (least harmonic majorant) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 9 E 2 (Ω) (Hardy-Smirnov class) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 11 dm (normalized Lebesgue measure on T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 12 ωz0 (harmonic measure) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 12 φG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 15 ψG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 16 f + , f − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.15 γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 16 b \ γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 16 C α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 16 D+ , D− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 17 φγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 18 fi , fe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 18 N(Ω) (Nevanlinna class of Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 21 N + (Ω) (Smirnov class of Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 21 H p (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 22 E p (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 22 Z(N) (common zeros of N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 25 k0N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 27 ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 27, p. 70 Pz (ζ ) (Poisson kernel of the disk) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 32 De (extended exterior disk) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 40 H02 (De ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 40 N0+ (De ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 40 Kzϑ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 47 S∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 47 g K zϑ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 49

xii

List of Symbols

Ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 59 Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 60 Gε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 68 kλM0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 70 [ f ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 80 M(ρ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 80 σe (essential spectrum) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 85

Preamble The statement of our main results, as well as the techniques used to prove them, become more meaningful if we review the basics of the Hardy space of the open unit disk D. The reader familiar with this material can skip this section. Several standard texts are [32, 35, 46, 52, 58]. For 0 < p < ∞, let H p , the Hardy space, denote the space of functions f analytic on D for which the L p integral means M p (r; f ) :=

Z

2π

0

| f (reiθ )| p

dθ 2π

1/p

remain bounded as r ↑ 1− . This definition can be extended to p = ∞ by M∞ (r; f ) := sup{| f (reiθ )| : θ ∈ [0, 2π ]} and so H ∞ is the set of bounded analytic functions on D. The function r 7→ M p (r; f ) is increasing on the interval [0, 1) and the quantity k f kH p := sup M p (r; f ) = lim M p (r; f ) r↑1−

00 are those of g. Here is a collection of standard facts about H p . Proofs of the results below, as well as the rest of the material from this chapter, are found in [32]. Theorem. For 0 < p 6 ∞ and f ∈ H p , 1.

f (eiθ ) := lim f (reiθ ) r→1−

exists for almost every θ . 2. This almost everywhere defined boundary function eiθ 7→ f (eiθ ) belongs to L p and when 0 < p < ∞, Z 2π dθ | f (reiθ ) − f (eiθ )| p = 0. lim 2π r→1− 0 Hence k f kH p = k f kL p .

3. If f ∈ H p \ {0}, then

Z 2π 0

log | f (eiθ )|

dθ > −∞ 2π

and hence the function eiθ 7→ f (eiθ ) can not vanish on any set of positive measure.

4. If p > 1, and f ∈ H p has Taylor series

∞

f (z) =

∑ an zn ,

n=0

then an =

Z 2π 0

f (eiθ )e−inθ

dθ , n ∈ N0 . 2π

5. If p > 1 and f ∈ H p , we have the Cauchy integral formula f (z) =

1 2π i

I

f (ζ ) dζ , ζ T −z

where T := ∂ D. 6. For 0 < p < ∞, the polynomials are dense in H p . When p = ∞, the polynomials are weak-∗ dense1 in H ∞ . From our collection of facts about H 2 , one can show that the inner product on H 2 can be written as Z 2π dθ h f , gi = f (eiθ )g(eiθ ) . 2π 0

Theorem (Smirnov). If 0 < p < q and f ∈ H p has Lq boundary values, then f ∈ H q . 1

See [35, p. 85] for more on the weak-∗ topology on H ∞ .

xv We know that, via boundary functions, H p can be viewed as a closed subspace of Turning this problem around, one can ask: when does a given f ∈ L p belong to H p ? At least for p > 1, there is an answer given by a theorem of F. and M. Riesz. L p.

Theorem. For p > 1, a function f ∈ L p belongs to H p if and only if the Fourier coefficients Z 2π dθ f (eiθ )e−inθ 2π 0 vanish for all n < 0. This result generalized to measures. Theorem (F. and M. Riesz theorem). Suppose a finite complex Borel measure µ on T satisfies Z 2π einθ d µ (eiθ ) = 0 ∀n ∈ N0 . 0

Then d µ = φ

dθ 2π

where φ ∈ H01 = { f ∈ H 1 : f (0) = 0}.

Every f ∈ H p can be factored as

f = Of If . The function O f , the outer factor, is characterized by the property that O f belongs to H p and Z 2π dθ log |O f (eiθ )| . log |O f (0)| = 2π 0 Every H p outer function F (i.e., F has no inner factor) can be expressed as Z 2π iθ e +z iθ d θ F(z) = eiγ exp log ψ (e ) , 2π eiθ − z 0 where γ is a real number, ψ > 0, log ψ ∈ L1 , and ψ ∈ L p . Note that F has no zeros in the open unit disk and |F(eiθ )| = ψ (eiθ ) almost everywhere. Moreover, every such F as above belongs to H p and is outer. The inner factor, I f , is characterized by the property that I f is a bounded analytic function on D whose boundary values satisfy |I f (eiθ )| = 1 for almost every θ . Furthermore, the inner factor I f can be factored further as the product of two inner functions I f = bsµ , where b is a Blaschke product ∞

|an | an − z n=1 an 1 − an z

b(z) = zm ∏

whose zeros at z = 0 as well as {an } ⊂ D\{0} (repeated according to multiplicity) satisfy the Blaschke condition ∞

∑ (1 − |an|) < ∞,

n=1

Preamble

xvi

(which guarantees the convergence of the product) and sµ is the (zero free) singular inner factor Z 2π iθ e +z iθ d µ (e ) , sµ (z) = exp − eiθ − z 0

where µ is a positive measure on T which is singular with respect to Lebesgue measure. A meromorphic function f on D is said to be of bounded type if f = h1 /h2 , where h1 , h2 are bounded analytic functions on D. From our above discussion, a function of bounded type must have finite non-tangential limits almost everywhere on T and can be factored as Ih Oh f = 1 1. Ih2 Oh2

The set N, the Nevanlinna class, will be the functions f of bounded type which are analytic on D (equivalently Ih2 is a singular inner function). The set N + , the Smirnov class, will be the set of f ∈ N for which Ih2 is a constant. This extension of Smirnov’s theorem (see above) due to Polubarinova and Kochina [58, p. 80] (see also [32, p. 28]), will be used many times in this book. Theorem. If f ∈ N + with L p boundary function, then f ∈ H p . It turns out that functions in H p not only have a radial limit almost everywhere on T but also have a stronger non-tangential limit almost everywhere. We collect some facts about non-tangential limits which will be used at various times in the text. For ζ ∈ T and α > 1, let Γα (ζ ) := {z ∈ D : |z − ζ | < α (1 − |z|)} be a non-tangential approach region (often called a Stoltz region). Note that Γα (ζ ) is a triangular shaped region with its vertex at ζ (see Fig. 1).

ζ

Γα (ζ)

Figure 1: Non-tangential approach region with vertex at ζ ∈ T

xvii We say that f has a non-tangential limit value A at ζ , written ∠ lim f (z) = A, z→ζ

if f (z) → A as z → ζ within any non-tangential approach region Γα (ζ ).

Theorem. If f ∈ H p , 0 < p 6 ∞, then f has a finite non-tangential limit almost everywhere on T. From our collection of facts about H p functions, we know that functions in H p \ {0} can not have non-tangential limit equal to zero on an any set of positive measure. This fact is not unique to H p functions. Theorem (Privalov’s uniqueness theorem [17, 52, 58]). Suppose f is analytic on D and ∠ lim f (z) = 0 z→ζ

for ζ in some subset of T of positive Lebesgue measure. Then f ≡ 0. Non-tangential limits are important in the statement of Privalov’s theorem since there are non-trivial analytic functions on D which have radial limits equal to zero almost everywhere on T [12]. Since invariant subspaces is the heart of this book, let us mention Beurling’s theorem. The shift operator S : H 2 7→ H 2 , defined by (S f )(z) = z f (z), is an isometry on H 2 . A classical theorem of A. Beurling [13] (see also [32]) characterizes the invariant subspaces of S. By ‘invariant subspace’ we mean a closed linear manifold M ⊂ H 2 for which SM ⊂ M. If ϑ is an inner function, then kϑ f k = k f k for all f ∈ H 2 and so ϑ H 2 is a closed linear manifold (a subspace) of H 2 . It is also clearly S-invariant. Beurling’s theorem says these are all of them. Theorem (Beurling). If ϑ is an inner function, the set ϑ H 2 is an S-invariant subspace of H 2 . Conversely, if M ⊂ H 2 , M 6= {0}, is an S-invariant subspace, then M = ϑ H 2 for some inner function ϑ . Since the main purpose of this book is to essentially prove a version of Beurling’s theorem for the Hardy space of a slit disk, we include a proof of Beurling’s theorem for the disk. Proof of Beurling’s theorem. The proof that ϑ H 2 is an S-invariant subspace of H 2 was discussed in our preliminary remarks. To prove the second part of the theorem, suppose M is a non-zero S-invariant subspace of H 2 . First notice that SM 6= M. If this were not the case, then f /z ∈ M whenever f ∈ M. Applying this k times we conclude that f ∈ M ∀k ∈ N. zk

Preamble

xviii

But this would mean, since f /zk must be analytic on D, that f ≡ 0, a contradiction to the assumption that M 6= {0}. Second, since S is an isometry, SM is closed and since SM 6= M, one observes that M ∩ (SM)⊥ 6= {0}. Thus M ∩ (SM)⊥ contains a non-trivial function ϑ . We now argue that |ϑ | = c on a set of full measure in T. Indeed, Z 2π 0

|ϑ (eiθ )|2 e−inθ

dθ = hϑ , Sn ϑ i = 0 2π

∀ n ∈ N.

Taking complex conjugates of both sides of the above equation, we also see that Z

T

|ϑ (eiθ )|2 einθ

dθ = 0 ∀ n ∈ N. 2π

This means that the Fourier coefficients of |ϑ |2 all vanish except for n = 0 and so |ϑ |2 = c almost everywhere on T. Without loss of generality, we can assume that |ϑ | = 1 almost everywhere on T and so ϑ is an inner function. Third, let [ϑ ] denote the closed linear span of the functions

ϑ , Sϑ , S2 ϑ , · · · and observe that [ϑ ] = ϑ H 2 . To see this, notice that clearly [ϑ ] ⊂ ϑ H 2 . For the other containment, let g = ϑ G ∈ ϑ H 2 and let GN be the N-th partial sum of the Taylor series of G. Notice that ϑ GN ∈ [ϑ ] since GN is a polynomial. From Parseval’s theorem, GN → G in H 2 and so, since ϑ is a bounded function, ϑ GN converges to ϑ G in H 2 . Finally, observe that [ϑ ] = M. Indeed, ϑ ∈ M and so [ϑ ] ⊂ M. Now suppose that f ∈ M and f ⊥ [ϑ ]. Since f ⊥ [ϑ ], Z 2π 0

f (eiθ )ϑ (eiθ )einθ

dθ = h f , Sn ϑ i = 0 2π

∀n ∈ N0 .

dθ = hSn f , ϑ i = 0 2π

∀n ∈ N.

But since ϑ ⊥ SM, we also know that Z 2π 0

f (eiθ )ϑ (eiθ )einθ

The previous two equations say that all of the Fourier coefficients of f ϑ vanish and so f ϑ = 0 almost everywhere on T. But we have already shown that |ϑ | = 1 almost everywhere on T and so f ≡ 0.

xix The key to proving Beurling’s theorem is the fact that the invariant subspace generated by M ∩ (SM)⊥ is equal to M. This idea extends to other Hilbert spaces of analytic functions [8, 61, 69], but not to the Hardy space of a slit domain. There is a Beurling theorem for the H p spaces [32, 35]: suppose 0 < p < ∞ and M is a non-zero subspace of H p . Then M is S-invariant if and only if M = ϑ H p for some inner function ϑ .

Chapter 1

Introduction 1.1 Some history This monograph continues the study of the invariant subspaces of the Hardy space H 2 (Ω) of a bounded domain Ω ⊂ C (see Chapter 2 for a definition of the Hardy space). By the term ‘invariant subspace’, we mean a subspace (i.e., a closed linear manifold) M of H 2 (Ω) for which SM ⊂ M, where (S f )(z) = z f (z) is the operator ‘multiplication by z’. When Ω is the open unit disk D := {z ∈ C : |z| < 1}, a much celebrated theorem of Beurling ([23, p. 135] or [35, p. 82]) says that every non-trivial invariant subspace M takes the form M = ΘH 2 (D), where Θ is an inner function meaning that Θ is a bounded analytic function on D whose non-tangential boundary function has constant modulus one almost everywhere on ∂ D. A similar result is true when Ω is a Jordan domain with smooth boundary [33]. When Ω is a finitely connected (bounded) domain with disjoint analytic boundary contours - an annulus for example - the invariant subspaces of H 2 (Ω) were examined in a series of papers beginning with Sarason [66] and continuing with Hasumi [38], Voichick [77, 78], Royden [65], Hitt [45], Yakubovich [79], Aleman and Richter [7], Aleman and Olin [6]. Due to the ‘holes’ in the region, there are several types of invariant subspaces to consider. First there are the ‘fully invariant’ subspaces M which, by definition, satisfy rM ⊂ M for every rational function r whose poles are off of Ω− . In this case [38, 66, 77, 78], M = ΘH 2 (Ω), where Θ is an inner function on Ω meaning that Θ is a bounded analytic function on Ω whose non-tangential boundary function has constant modulus on each of the components of ∂ Ω. Then there are the invariant subspaces M for which rM ⊂ M for rational functions r whose poles lie in certain components of C \ Ω− but not others. In this case [7, 45, 65, 79], the description of these M depends on, amongst other things, the behavior of the pseudocontinuation of functions across the holes. The purpose of this monograph is to broaden the discussion to include (simply connected) slit domains, Ω := D \

N [

j=1

γ j,

(1.1.1)

Chapter 1. Introduction

2

where γ1 , . . . , γN are simple disjoint analytic arcs (see Chapter 10 for the precise technical restrictions on the arcs). See Figure 1.1 for an example.

γ2 γ3

γ1 γ4

Figure 1.1: A (simply connected) slit domain Ω. For example, perhaps the simplest type of slit domain to consider here is G = D \ [0, 1] (see Figure 1.2).

1 0

Figure 1.2: The slit domain G = D \ [0, 1).

1.2 Invariant subspaces of the slit disk Before stating one of our main theorems, the complete description of the invariant subspaces of H 2 (Ω) for slit domains, let us give some examples. For clearer exposition, we state these examples, as well as our main results, for the simple slit domain G. The results for the more general slit domains Ω in (1.1.1) are stated in Chapter 10. The description of the invariant subspaces of H 2 (G) depends on, amongst other things, the behavior of the functions near the boundary of G. Here is where we encounter

1.2. Invariant subspaces of the slit disk

3

our first major difference between the case discussed previously, where the domain was an annular type domain, and our current case of the slit domain G. In the former case, when computing the boundary function, the boundary is accessible from only one direction. In the latter case however, part of the boundary - the slit - is accessible from two directions - from the top and from the bottom of the slit. For each f ∈ H 2 (G), the functions f + (x) := lim f (x + iy) and y→0+

f − (x) := lim f (x + iy) y→0−

are defined for almost every x ∈ [0, 1] and f + , f − ∈ L2 ([0, 1], ω ), where ω is harmonic measure for G. Furthermore, f (eiθ ) := lim f (reiθ ) r→1−

exists for almost every θ and this function belongs to L2 (T, ω ), where T = ∂ D. In fact (Proposition 2.3.4), the norm on H 2 (G) satisfies k f k2H 2 (G) =

Z

[0,1]

Z | f + |2 + | f − |2 d ω + | f |2 d ω . T

We will discuss these preliminaries in Chapter 2.

The first type of invariant subspaces M ⊂ H 2 (G) to consider are those for which ⊂ M, where H ∞ (G) is the set of bounded analytic functions on G. By realizing

H ∞ (G)M that

H 2 (G) = f ◦ φG−1 : f ∈ H 2 (D) ,

where φG is a conformal map from D onto G (see the Appendix for a specific formula), and using Beurling’s theorem, we can characterize these H ∞ (G)-invariant subspaces by means of ‘inner’ functions. We say a bounded analytic function Θ on G is G-inner if Θ ◦ φG is inner in the classical sense (the almost everywhere defined boundary function on ∂ D has constant modulus one almost everywhere), or equivalently, the boundary function defined by Θ+ , Θ− , and Θ|T, as above, has modulus one almost everywhere. These H ∞ (G)-invariant subspaces are characterized in Proposition 6.1.2 as follows. Theorem 1.2.1. Suppose M is a non-trivial subspace of H 2 (G) such that H ∞ (G)M ⊂ M.

(1.2.2)

Then there is a G-inner function Θ such that M = ΘH 2 (G). Moreover, for every G-inner function Θ, the subspace M := ΘH 2 (G) satisfies (1.2.2). of

The subspaces ΘH 2 (G), where Θ is G-inner, are not all of the invariant subspaces To see this, we need only consider the class of invariant subspaces M(ρ , E) := f ∈ H 2 (G) : f + = ρ f − a.e. on E ,

H 2 (G).

where ρ : [0, 1] → C is a measurable function and E is a measurable subset of [0, 1]. Since convergence of a sequence in the H 2 (G)-norm implies a subsequence converges


4

almost everywhere on ∂ G, we see that M(ρ , E) is closed in H 2 (G) and, since the identity function z 7→ z is analytic across the slit [0, 1], M(ρ , E) is clearly invariant. The subspace M(ρ , E) is never equal to ΘH 2 (G) for some G-inner function Θ since M(ρ , E) can not √ contain both Θ and Θ z. Perhaps every invariant subspace takes the form ΘM(ρ , E). Though this seems reasonable, it is not the case. Consider the functions from H 2 (G) which have an analytic continuation across [0, 1). We will show in Corollary 6.1.8 that this class of functions is closed in H 2 (G) and is equal to M(1, [0, 1]). In fact, see (7.3.1) and Theorem 7.3.2, the unit ball in this space forms a normal family on D. Hence M0 := { f ∈ M(1, [0, 1]) : f (0) = 0} is closed in H 2 (G) and is an invariant subspace of H 2 (G). Moreover, since the common zero of M0 is at the origin, which is on slit and not in G (where one could take it into account using the G-inner function Θ), this space is not of the form ΘM(ρ , E). Note that M0 can be equivalently described as f f ∈ M(1, [0, 1]) : ∈ H 2 (G) . z Furthermore, the function F(z) = z is G-outer in that F ◦ φG is outer in the classical sense of Z |d ζ | log |F ◦ φG (0)| = log |F ◦ φG (ζ )| . 2π T These examples indicate that the description of the invariant subspaces depends on the four parameters Θ, ρ , E, F. Our main theorem (Corollary 6.1.6 and Theorem 6.2.1) codifies this as follows: For ε ∈ (0, 1), let Gε := D \ [−ε , 1]. Theorem 1.2.3. Let M be a non-trivial invariant subspace of H 2 (G) with greatest common G-inner divisor ΘM . Then there exists a measurable set E ⊂ [0, 1], a measurable function ρ : [0, 1] → C and, given any ε ∈ (0, 1), there exists a Gε -outer function Fε such that f 2 + − 2 M = ΘM · f ∈ H (G) : ∈ H (Gε ), f = ρ f a.e. on E . (1.2.4) Fε An alert reader might wonder why such a linear manifold defined by the righthand side of (1.2.4) is actually closed, i.e., a subspace. Theorem 1.2.3 does not say that for any Gε -outer functions Fε the linear manifold in (1.2.4) is closed. It just says that given M there are some Gε -outer functions Fε for which M can be written in terms of these particular Fε ’s (and the other parameters ΘM , ρ , and E). Furthermore, the Fε ’s are not unique. The set in (1.2.4) remains unchanged if Fε is replaced by Fε F, where F is any G-outer function satisfying 0 < δ1 6 |F| 6 δ2 < ∞. This somewhat mysterious outer function also appears in the description of the invariant subspaces of the Hardy space of a multiply connected domain [6, 7, 45, 79]. Although the Fε ’s are not unique, we will show in Corollary 3.6.3 that the other parameters ΘM , ρ , and E, are (essentially) unique.

1.3. Nearly invariant subspaces

5

1.3 Nearly invariant subspaces The main tool in proving Theorem 1.2.3, which turns out to be interesting in its own right, is the concept of a ‘nearly invariant’ subspace first explored by Sarason [68] and Hitt [45]. Indeed, as we shall see in Corollary 6.1.3, if M is an invariant subspace of H 2 (G), then N := f ◦ α −1 : f ∈ M ,

b \ γ (see (2.3.8) below), is a nearly where α is a particular conformal map from G onto C 2 b \ γ ). Here C b = C ∪ {∞} is the Riemann sphere and invariant subspace of H (C n o π γ = α (∂ G) = eit : − 6 t 6 π . 2 b γ ), for which the origin is not a common zero for functions We say a subspace N ⊂ H 2 (C\ in N, is nearly invariant if f /z ∈ N whenever f ∈ N and f (0) = 0. Nearly invariant subspaces have been, and continue to be, explored in various settings [5, 7, 40, 45, 55, 68]. In fact, very recently, nearly invariant subspaces have even appeared in mathematical physics [51]. If ϕ is the normalized reproducing kernel at the origin for N, we will show in Corollary 3.3.1 that the operator fi ϕe J : N → H 2 (D) ⊕ L2 (γ , ω ), J f = , fe − fi ϕi ϕi b \ γ and, for f ∈ H 2 (C b \ γ ), is an isometry. Here ω is harmonic measure for C fi := f |D,

fe := f |De .

In the expression,

ϕe ϕi in the second component in the definition of J f , we are using the appropriate almost everywhere defined boundary functions, i.e., fe − fi

fe (ζ ) := lim f (rζ ), r→1+

fi (ζ ) := lim f (rζ ), r→1−

ζ ∈ γ.

If Mζ denotes the operator on H 2 (D) ⊕ L2 (γ , ω ) defined by multiplication by the independent variable on each component function, then (JN)⊥ becomes an invariant subspace for Mζ and we use the Wold decomposition for Mζ |(JN)⊥ to describe (JN)⊥ . We then use annihilators to describe N (Theorem 3.1.2). Our description of the nearly invarib \ γ ) is the following. ant subspaces of H 2 (C

b γ ) with greatTheorem 1.3.1. Let N be a non-trivial nearly invariant subspace of H 2 (C\ b \ γ -inner divisor ΘN . Then there exists a D-outer function F, a measurable est common C set E ⊂ γ , and a measurable function ρ : γ → C such that b \ γ ) : fi ∈ H 2 (D), fi = ρ fe a.e. on E . N = ΘN · f ∈ H 2 (C F


6

b \ γ ) in terms In Theorem 4.2.5 we describe the nearly invariant subspaces of H 2 (C of the invariant subspaces of the backward shift operator S∗ f :=

f − f (0) z

b \ γ ) in on H 2 (D). In Theorem 5.2.3 we describe the nearly invariant subspaces of H 2 (C b \ T. terms of de Branges-type spaces on C

1.4 Cyclic invariant subspaces

We also study the cyclic invariant subspaces of H 2 (G). By this we mean those invariant subspaces which take the form [ f ] := W

_

{Sn f : n ∈ N0 } .

Here denotes the closed linear span in H 2 (G) and N0 = N ∪ {0}. Not every invariant subspace of H 2 (G) is cyclic. In fact, H 2 (G) is not a cyclic subspace [4, Cor. 3.3] (see Example 8.2.13 for other examples). We have the following result about cyclic invariant subspaces. Theorem 1.4.1. If E ⊂ [0, 1] is the measurable set corresponding to the (non-zero) invariant subspace M from Theorem 1.2.3 and [0, 1] \ E has positive measure then M is not cyclic. Though not every M is cyclic, we will show in Theorem 7.1.1 that every M is 2cyclic. Theorem 1.4.2. For an invariant subspace M of H 2 (G), there are two functions f , g ∈ M so that _ M = {zm f , zn g : m, n ∈ N0 } .

In fact, one can take f and g to be certain ‘extremal’ functions for M. In Theorem 7.1.2 we will determine, for general f , g ∈ H 2 (G), when the invariant subspace generated by f and g is all of H 2 (G). Theorem 1.4.3. If f , g ∈ H 2 (G) \ {0}, then _

{zm f , zn g : m, n ∈ N0 } = H 2 (G)

if and only if f and g have no common G-inner factor and the set f + (x) g+ (x) x ∈ [0, 1) : − = f (x) g− (x) has Lebesgue measure zero.

1.5. Essential spectrum

7

Though not every invariant subspace of H 2 (G) is cyclic, we can, under certain circumstances, compute the cyclic invariant subspace [ f ] (see Theorem 7.2.2). Theorem 1.4.4. Suppose that both f and 1/ f belong to H 2 (G) and ρ := f + / f − . Then [ f ] = M(ρ , [0, 1]). The equality [ f ] = M(ρ , [0, 1]) is not true in general (see Example 7.2.4). In Corollary 6.1.3 we will show that every invariant subspace of H 2 (G) is nearly invariant. This implies, assuming M 6= {0}, that for any λ ∈ G, dim(M ⊖ (z − λ )M) = 1. However, the analogue of Beurling’s Theorem1 does not hold in the slit disk, in the sense that M⊖(z− λ )M does not always generate M. This is simply because not every invariant subspace of H 2 (G) (H 2 (G) in fact!) is cyclic (see also Example 8.2.13).

1.5 Essential spectrum It is known that if S f = z f on H 2 (G), then σ (S), the spectrum of S, is equal to G− = D− [21, Prop. 4.1] and that σe (S), the essential spectrum of S, is equal to ∂ G [21, Thm. 4.3]. In Theorem 8.2.5, we compute the essential spectrum of S|M, where M is an invariant subspace of H 2 (G). Theorem 1.5.1. Let M be a non-zero invariant subspace of H 2 (G) and let A(M) be the set of points x ∈ [0, 1) with the property that there exists an fx ∈ M such that f / fx extends to be analytic in a neighborhood of x whenever f ∈ M. Then we have

σe (S|M) = ∂ G \ A(M). Although every cyclic invariant subspace is contained in M(ρ , [0, 1]) for some ρ and Theorem 1.4.4 says that certain cyclic subspaces are of the form M(ρ , [0, 1]), Theorem 1.5.1 enables us to do the following (see Example 8.2.13). Theorem 1.5.2. There are measurable functions ρ : [0, 1] → C for which the invariant subspace M(ρ , [0, 1]) is not cyclic. As mentioned earlier, our results have analogs when the slit domain G = D \ [0, 1] is replaced by a slit domain of the form in (1.1.1) (Theorem 10.1.2). Our results also have analogs when the Hardy space H 2 (G) is replaced by the Hardy-Smirnov space E 2 (G) (see (11.0.1)). Finally, we mention that in Theorem 7.3.2 we apply our main theorems to describe P2 (ω ), where ω is harmonic measure on G = D \ [0, 1) and P2 (ω ) is the closure of the analytic polynomials in L2 (ω ). Along the way, we compute the set of bounded point evaluations for P2 (ω ).

1 If

one looks at the proof of Beurling’s theorem from the Preamble, one can see, for a (non-zero) invariant subspace M of H 2 (D), that dim(M ⊖ zM) = 1 and that the invariant subspace generated by M ⊖ zM is M.

Chapter 2

Preliminaries 2.1 Hardy space of a general domain In this chapter, we set our notation and review some elementary facts about the Hardy spaces of general (simply connected) domains. Some good references for this material b := C ∪ {∞}, we say are [21, 23, 24, 32, 33]. For a simply connected domain Ω ⊂ C that an upper semicontinuous function u : Ω → [−∞, ∞) is subharmonic if it satisfies the sub-mean value property. That is to say, at each point a ∈ Ω, there is an r > 0 so that u(a) 6

Z 2π

u(a + reiθ )

0

dθ . 2π

(2.1.1)

If f is analytic on Ω and p > 0, then | f | p is subharmonic. We say a subharmonic function u has a harmonic majorant if there is a harmonic function U on Ω such that u 6 U on Ω. By the Perron process for solving the classical Dirichlet problem [10, p. 200] [59, p. 118], one can show that if a subharmonic function u 6≡ −∞ has a harmonic majorant, then u has a least harmonic majorant U in that u 6 U 6 V on Ω for all harmonic majorants V of u. We say an analytic function f on Ω belongs to the Hardy space H 2 (Ω) if the subharmonic function | f |2 has a harmonic majorant in Ω. If z0 ∈ Ω, we can norm H 2 (Ω) by q k f kH 2 (Ω) =

U f (z0 ),

(2.1.2)

where U f is the least harmonic majorant for | f |2 . By the mean value property for harmonic functions (i.e., equality in (2.1.1)1 ), notice that either U f > 0 or U f ≡ 0 on Ω. Thus k f kH 2 (Ω) actually defines a norm on H 2 (Ω). Furthermore, we can use Harnack’s inequality2 to show that different norming points z0 yield equivalent norms on H 2 (Ω). The following three simple facts will be used several times: fact [18, p. 260], u ∈ C(Ω) is harmonic on Ω if and only if u satisfies the mean value property on Ω. inequality: For fixed z1 ,z2 ∈ Ω there is a C > 0 so that C−1U(z1 ) 6 U(z2 ) 6 CU(z1 ) for every positive harmonic function U on Ω [59, p. 14]. 1 In

2 Harnark’s

Chapter 2. Preliminaries

10

b and φ is a conformal map 1. Suppose Ω1 , Ω2 are two simply connected domains in C from Ω1 onto Ω2 . If z0 ∈ Ω1 is the norming point for H 2 (Ω1 ) and φ (z0 ) is the norming point for H 2 (Ω2 ), the operator f 7→ f ◦ φ

(2.1.3)

is a unitary operator from H 2 (Ω2 ) onto H 2 (Ω1 ); 2. Let Ω1 ⊂ Ω2 and z0 ∈ Ω1 . Suppose z0 is the norming point for both H 2 (Ω1 ) and H 2 (Ω2 ). If f ∈ H 2 (Ω2 ), then f ∈ H 2 (Ω1 ) and k f kH 2 (Ω1 ) 6 k f kH 2 (Ω2 ) ;

(2.1.4)

3. Given a compact set K ⊂ Ω, there is a positive constant CK , depending only on K, such that | f (z)| 6 CK k f kH 2 (Ω) , z ∈ K, f ∈ H 2 (Ω). (2.1.5) In certain cases (see Proposition 2.4.13 below) one can estimate the constant CK . The inequality in (2.1.5) says that H 2 (Ω), when endowed with the norm in (2.1.2), becomes a Hilbert space of analytic functions on Ω with inner product h f , gi = U f (z0 )Ug (z0 ). By this we mean that H 2 (Ω) is not only a Hilbert space comprised of analytic functions on Ω but the natural injection i : H 2 (Ω) → Hol(Ω) (the analytic functions on Ω endowed with the topology of uniform convergence on compact sets) is continuous. The inequality in (2.1.5) also says that for each fixed z ∈ Ω, the linear functional f 7→ f (z) is continuous and so, by the Riesz representation theorem, there is a kz ∈ H 2 (Ω) such that h f , kz i = f (z)

∀ f ∈ H 2 (Ω).

Thus H 2 (Ω) is a reproducing kernel Hilbert space. We will see in a moment that the inner product h f , gi can be represented as an integral.

When Ω is the open unit disk D, there is a more classical, but equivalent, definition of H 2 (D) in terms of integral means. Indeed, as mentioned in the Preamble at the beginning of this book, an analytic function f on D belongs to H 2 (D) if and only if sup

Z 2π

00

is a reducing subspace for Mζ , and Mζ |H1 is a unilateral shift.

Proposition 3.4.2. There is a measurable set E = E(N) ⊂ γ such that H0 = {0} ⊕ χE L2 (γ , ω ).

Moreover, if F ⊂ γ and

JN ⊂ H 2 ⊕ χF c L2 (γ , ω ),

then m(F \ E) = 0. Proof. To see the first part, recall that Mζ = S ⊕ T , where (S f )(z) = z f (z), and

f ∈ H 2 := H 2 (D)

(T g)(ζ ) = ζ g(ζ ),

g ∈ L2 (γ , ω ).

Chapter 3. Nearly invariant subspaces

38 Since

\

n>0

we see that H0 =

\

Sn H 2 = {0},

(S ⊕ T )n H = {0} ⊕ Y,

n>0

where Y is a T -invariant subspace of L2 (γ , ω ). Furthermore, since γ 6= T, we know that both ζ Y ⊂ Y and ζ Y ⊂ Y hold (use Lavrentiev’s theorem [23, p. 232] to approximate the function v(ζ ) = ζ uniformly on γ by a sequence of analytic polynomials). Now apply a classical theorem of Wiener [44, p. 7] to get Y = χE L2 (γ , ω ) for some measurable subset E ⊂ T. To see the second part of the proposition, observe that a routine argument shows that (H 2 ⊕ χF c L2 (γ , ω ))⊥ = {0} ⊕ χF L2 (γ , ω ). Thus, if we assume that JN ⊂ H 2 ⊕ χF c L2 (γ , ω ), we have H = (JN)⊥ ⊃ {0} ⊕ χF L2 (γ , ω ) and so H0 =

\

n>0

Mζn H ⊃ {0} ⊕ χF L2 (γ , ω ).

But from the first part of the theorem, H0 = {0} ⊕ χE L2 (γ , ω )

and the result follows.

Lemma 3.4.3. Let H0 = {0} ⊕ χE L2 (γ , ω ) as in Proposition 3.4.2. Then any φ = (a, b) ∈ H1 ⊖ Mζ H1 of unit norm, must satisfy a ∈ H ∞ (D); |a|2 + |b|2w = 1

a.e. on T4 ;

b = 0 a.e. on E; a = bwϕe 4

a.e on γ \ E.

We extend the domain of b to be all of T by defining it to be zero on T \ γ .

(3.4.4) (3.4.5) (3.4.6) (3.4.7)

3.4. The Wold decomposition

39

Proof. Let φ = (a, b) ∈ H1 ⊖ Mζ H1 of unit norm. Extend the domain of b to include the entire unit circle by setting it to be zero on T \ γ . From here it follows that for each n ∈ N, Z E D n 0 = (a, b), Mζn (a, b) 2 2 = (|a|2 + |b|2w)ζ dm(ζ ). H ⊕L (γ ,ω )

T

Taking complex conjugates, we see that all of the non-zero Fourier coefficients of |a|2 + |b|2 w are equal to zero and so this function is a (non-negative) constant. Since φ has unit norm, this constant must be equal to one. This proves (3.4.5) as well as the fact that a ∈ H ∞ (D) (since it is an H 2 function with bounded boundary values). Since φ = (a, b) ∈ H1 = (H0 )⊥ = ({0} ⊕ χE L2 (γ , ω ))⊥ (Proposition 3.4.2), we know that Z bχE gwdm = 0 for all g ∈ L2 (γ , ω ).

From basic measure theory, it follows that b = 0 almost everywhere on E, proving (3.4.6). By Lemma 3.3.5 we know that for every f ∈ N, fi (a − bwϕe ) fe − ϕe = 0 a.e. on γ . ϕi If F ⊂ γ \ E has positive measure, the second statement in Proposition 3.4.2 says that there is an f ∈ N so that fi fe − ϕe ϕi is non-zero on some subset F ′ of F of positive measure. It follows that a − bwϕe = 0 almost everywhere on F ′ and hence zero almost everywhere on γ \ E. This proves (3.4.7). Proposition 3.4.8. Mζ |H1 is a unilateral shift of multiplicity one.

Proof. From our earlier discussion of multiplicities of shifts, we need to show that H1 ⊖ Mζ H1 is one dimensional. Let φ j = (a j , b j ), j = 1, 2, belong to H1 ⊖ Mζ H1 with φ1 ⊥ φ2 . Recall from Lemma 3.4.3 that a j , b j , j = 1, 2, satisfy the properties (3.4.6) and (3.4.7). Since φ j ⊥ Mζn H1 for all n ∈ N and φ1 ⊥ φ2 we get D E φ1 , Mζn φ2 2 2 = 0 ∀n ∈ N0 , D

H ⊕L (γ ,ω )

φ2 , Mζn φ1

The first equation says that Z

T

E

H 2 ⊕L2 (γ ,ω )

n

n

n

n

=0

∀n ∈ N0 .

(a1 a2 ζ + b1b2 ζ w)dm = 0 ∀n ∈ N0 .

The second equation says that Z

T

(a2 a1 ζ + b2b1 ζ w)dm = 0 ∀n ∈ N0 .


40

The complex conjugate of the above equation, together with one just before it, say that all of the Fourier coefficients of the function a1 a2 + b1b2 w vanish. Thus a1 a2 + b1 b2 w = 0 a.e. on T. This means that either a1 or a2 must vanish identically (since we are assuming, as in Lemma 3.4.3, that we have extended the domains of b1 , b2 to all of T by making them zero off γ ) and so by (3.4.6) and (3.4.7), either φ1 or φ2 must vanish identically. The operator Mζ |H1 is a shift of multiplicity one (Proposition 3.4.8) and so if φ ∈ H1 ⊖ Mζ H1 and has norm one, then φ = (a, b) satisfies the properties of Lemma 3.4.3 as well as, via the consequences of the Wold decomposition in (3.4.1), H1 =

∞ _

Mζn φ .5

(3.4.9)

n=0

The function a satisfies one more property. But first we need a few definitions. b \ D− , and define Recall that De := C H02 (De ) := { f ∈ H 2 (De ) : f (∞) = 0},

N0+ (De ) := { f ∈ N + (De ) : f (∞) = 0}.

The space N + (De ), the Smirnov class, was defined in (2.4.4).

Lemma 3.4.10. If Θe denotes the greatest common De -inner factor of { fe : f ∈ N}, the function a(1/z)Θe (z) z 7→ , z ∈ De , ϕe (z) belongs to N0+ (De ). Proof. Using the fact that H1 ⊂ (JN)⊥ and (3.4.9), we see, for all f ∈ N that f fi n n , az + fe − ϕe , bζ = 0 ∀n ∈ N0 . ϕ ϕi H2 L2 (γ ,ω ) Write the above identity out as an integral and use the F. and M. Riesz theorem6 [32, p. 41] to see that fi fi a + fe − ϕe wbχγ ∈ H02 . ϕi ϕi More precisely, the above function is equal to h almost everywhere on T, where h ∈ H02 . 5 Here we are using the general fact that if V is a shift of multiplicity n on a Hilbert space X and {e : 1 6 j j 6 n} is a basis for X ⊖ (V X), then {V k e j : k ∈ N0 ,1 6 j 6 n} is a basis for X. See [25, Prop. 23.10]. 6 F. and M. Riesz theorem: If µ is a finite complex measure on the unit circle T whose Fourier coefficients b (n) vanish for all n ∈ N, then d µ = f dm for some f ∈ H 1 . µ

3.4. The Wold decomposition

41

On T \ γ , h=

fi fe a= a ϕi ϕe

z 7→

fe (z) a(1/z) ϕe (z)

and the function (3.4.11)

belongs to the Nevanlinna class of De . Since h(1/z) ∈ H02 (De ), and has the same nontangential boundary values as the function in (3.4.11) on T \ γ , these two functions must be identical (Privalov’s uniqueness theorem - see [52, p. 62]). Since the function in (3.4.11) belongs to H02 (De ) for all f ∈ N and Θe is the greatest common De -inner factor of { fe : f ∈ N}, the result now follows. The final technical lemma needed to prove Theorem 3.1.2 is the following. b \ γ ) with greatest common C b \ γ -inner Lemma 3.4.12. Suppose N is a subspace of H 2 (C divisor one. Then the greatest common D-inner divisor of N|D is equal to one. Proof. Let

∞ b N1 = closH 2 (C\ b γ ) (H (C \ γ )N)

b \ γ )N1 ⊂ N1 . Using the same proof as Proposition 6.1.2 below, we and note that H ∞ (C b \ γ -inner divisor of N is one, that N1 = H 2 (C b \ γ ). By see, since the greatest common C ∞ b \ γ ) and fn ∈ N so that the definition of N1 , there are φn ∈ H (C b \ γ ) as n → ∞. in H 2 (C

φn fn → χC\ b γ

It follows from (2.1.4) that

(φn fn )i → χD

in H 2 (D) as n → ∞.

Suppose that N|D has a non-constant greatest common D-inner divisor ϑ . Then the previous equation says that

ϑ

(φn fn )i −ϑ ϑ

→ 0 in the norm of L2 (m)

and consequently (φn fn )i →ϑ ϑ

in the norm of L2 (m).

But since (φn fn )i /ϑ ∈ H 2 (D) for all n, then ϑ ∈ H 2 (D), which is not the case. Thus, by contradiction, N|D has greatest common D-inner divisor one.


42

3.5 Proof of the main theorem We are now ready for the proof of Theorem 3.1.2. Proof of Theorem 3.1.2. Before getting to the crux of the proof, let us set up a few things and remind the reader what we have already shown. Without loss of generality, we can assume that ΘN ≡ 1. If this is not the case, apply the argument below to N/ΘN . Lemma 3.4.12 says that the greatest common D-inner factor of { fi : f ∈ N} is one as is the greatest common De -inner factor of { fe : f ∈ N}. Let E ⊂ γ be the measurable set from Proposition 3.4.2 and let F be the D-outer function which satisfies |F|2 = Let

|ϕi |2 1 + wχγ |ϕe |2

ρ :=

ϕi ϕe

a.e. on T.7

a.e. on γ .

(3.5.1)

(3.5.2)

Lemma 3.2.14 says that ( f /ϕ )i ∈ H 2 (D) whenever f ∈ N. Thus, since we are assuming that ΘN ≡ 1 (note the discussion in the second paragraph of the proof), it must be the case that ϕi is D-outer. (3.5.3) If a is the H ∞ (D) function from Lemma 3.4.3, we know, again since ΘN ≡ 1 (and the discussion in the second paragraph of the proof), from Lemma 3.4.10 that a(1/z) ∈ N0+ (De ). ϕe (z)

(3.5.4)

Our final reminder is that fi fi JN = , fe − ϕe : f ∈ N ⊂ H 2 ⊕ L2 (γ , ω ) ϕi ϕi and, from the proof of Proposition 3.4.2 and Lemma 3.2.14, JN ⊂ H 2 ⊕ χE c L2 (γ , ω ), that is to say, for all f ∈ N we have fi ∈ H 2, ϕi 7 Observe

that log

|ϕi |2 = log |ϕi |2 − log(1 + wχγ |ϕe |2 ) a.e. 1 + wχγ |ϕe |2

(3.5.5)

The first summand is integrable on T. Use the fact that ϕi ∈ H 2 (D) \ {0} (2.1.4) and (2.4.2) to see this. The second is integrable since |ϕe |2 w is integrable. Thus by standard Nevanlinna theory [32, Ch. 2], such a D-outer function F satisfying (3.5.1) exists.

3.5. Proof of the main theorem

43 fe −

and

fi ϕe ∈ L2 (γ , ω ), ϕi

fi ϕe = 0 a.e. on E. ϕi We are now at the crux of the proof. Let b \ γ ) : fi ∈ H 2 (D), fi = ρ fe a.e. on E . N1 := f ∈ H 2 (C F fe −

(3.5.6)

(3.5.7)

To show

N = N1 , we begin with the following claim. Claim 1: N ⊂ N1 .

To prove this claim, we suppose that f ∈ N and notice that F is D-outer and so fi /F ∈ N + (D). We now prove that fi /F ∈ H 2 (D) by showing it has L2 (m) boundary values. Indeed, by the definition of the D-outer function F in (3.5.1), Z 2 Z 2 fi dm = fi (1 + wχγ |ϕe |2 ) dm T F T ϕi Z 2 Z fi fi 2 = dm + γ ϕi ϕe w dm. T ϕi

From (3.5.5) and (3.5.6), both of these integrals converge. Thus fi /F belongs to H 2 (D). To finish the proof of Claim 1, we can use (3.5.7), to see fi ϕi = =ρ fe ϕe

a.e. on E.

Thus f ∈ N1 , which proves Claim 1.

Claim 2: N1 ⊂ N.

Let g ∈ N1 and define xg :=

gi gi , ge − ϕe . ϕi ϕi

We first want to show that xg ∈ H 2 ⊕ L2 (γ , ω ). By (3.5.3), gi ∈ N + (D). ϕi

To show this function belongs to H 2 (D), we need to show it has L2 (m) boundary values. Indeed, from (3.5.1), Z 2 Z 2 Z 2 1 gi gi dm = gi dm 6 dm. 1 + wχγ |ϕe |2 T ϕi T F T F


44

By the definition of N1 , we know that gi /F ∈ H 2 (D). Thus it follows that gi ∈ H 2 (D). ϕi Furthermore, to show ge −

(3.5.8)

gi ϕe ∈ L2 (γ , ω ), ϕi

b \ γ ) norm). notice that the first term belongs to L2 (γ , ω ) (from the definition of the H 2 (C For the second term, observe that 2 Z Z 2 Z 2 2 gi ϕe w dm = gi w|ϕe | dm 6 gi dm 1 + w|ϕe |2 γ ϕi γ F γ F which is finite since gi /F ∈ H 2 (D). Thus

xg ∈ H 2 ⊕ L2 (γ , ω ). We now want to show xg ∈ JN. We will do this by proving xg ⊥ (JN)⊥ .

Note here that J is an isometry and so JN is closed. Thus xg ∈ JN ⇔ xg ⊥ (JN)⊥ . Using Proposition 3.4.2 and (3.4.9) we have _ n (JN)⊥ = H0 ⊕ H1 = {0} ⊕ χE L2 (γ , ω ) ⊕ Mζ φ . n>0

By the definition of N1 , we know that ge − Hence,

gi ϕe = 0 a.e. on E. ϕi

(3.5.9)

xg ⊥ {0} ⊕ χE L2 (γ , ω ).

We are left with showing hxg , Mζn φ iH 2 ⊕L2 (γ ,ω ) = 0 ∀n ∈ N0 . If φ = (a, b) as in (3.4.9) (with the understanding that the domain of b is all of T be defining it to be zero on T \ γ ), the F. and M. Riesz theorem says that

xg , (zn a, ζ n b) H 2 ⊕L2 (γ ,ω ) = 0 ∀n ∈ N0 if and only if

gi gi a + ge − ϕe wb ∈ H02 . ϕi ϕi

(3.5.10)

3.5. Proof of the main theorem

45

By considering three cases: T \ γ , γ \ E, and E, and using (3.4.6) and (3.4.7), along with the facts that gi /ϕi = ge /ϕe on T \ γ and almost everywhere on E, one can show that the function on the left-hand side of (3.5.10) is equal to ge a a.e. on T. ϕe But by (3.5.4) we can prove that this function belongs to H02 (De ) by showing it as L2 (m) boundary values. Indeed, Z

2 Z Z Z ge a dm = + + . T ϕe E γ \E T\γ

(3.5.11)

For the first integral in (3.5.11),

Z

2 2 Z ge a dm = gi a dm. E ϕe E ϕi

This integral converges since a ∈ H ∞ (D) and (g/ϕ )i ∈ H 2 (D) (3.5.8). For the second integral in (3.5.11), Z

ge 2 a dm = γ \E ϕe =

=

2 ge bϕe w dm γ \E ϕe

Z

Z

γ \E

Z

γ \E

(by (3.4.7))

|ge |2 |b|2 w w dm |ge |2 (1 − |a|2)wdm

(by (3.4.5)).

b \ γ ) and the fact that The above integral converges by the definition of the norm on H 2 (C ∞ a ∈ ball(H ). The third integral in (3.5.11) converges since ge /ϕe analytically continues to gi /ϕi across T \ γ and so Z g e 2 gi 2 a dm = a dm T\γ ϕe T\γ ϕi

Z

which converges since (g/ϕ )i ∈ H 2 (D) (see (3.5.8)) and a ∈ H ∞ (D). Thus xg ∈ JN and so xg = J f for some f ∈ N. However, by the definition of the operator J, gi fi = ϕi ϕi and so g = f ∈ N. This proves Claim 2 and hence the proof.


46

3.6 Uniqueness of the parameters Let F be a closed subset of the arc γ . A well-known result of Ahlfors and Beurling [1] or b \ F) contains non-constant functions if and only [34] (see p. 6 and p. 29) says that H ∞ (C 8 if m(F) > 0. This means, via a version of Morera’s theorem [35, p. 95], that if m(F) > 0, b \ F) such that then there is an f ∈ H ∞ (C fe 6= 1 fi

a.e. on F ′ ,

(3.6.1)

for some compact F ′ ⊂ F with m(F ′ ) > 0. This observation along with Theorem 3.1.2 yields the following two corollaries. Corollary 3.6.2. Let A be a closed subset of γ . Then a nearly invariant subspace N of the b \ A)-invariant if and only if m(A ∩ E) = 0. form in (3.1.3) is H ∞ (C Corollary 3.6.3. Let N be a nearly invariant subspace of the form in (3.1.3).

b \ γ )-invariant, then the parameters Θ, E, and ρ are unique in the 1. If N is not H ∞ (C sense that if Θ j , E j , ρ j , j = 1, 2, represent N, then Θ1 = eit Θ2 , m(E1 ∆E2 ) = 0, and ρ1 χE1 = ρ2 χE2 almost everywhere. b \ γ )N ⊂ N, then Θ is unique up to a unimodular constant and m(E) = 0. 2. If H ∞ (C

b \ γ -inner divisor of the functions in N Proof. The parameter Θ is the greatest common C and thus is unique up to a unimodular constant. Suppose there are two subsets E1 , E2 of γ and functions ρ1 , ρ2 which represent the same nearly invariant subspace N as in (3.1.3) and m(E1 \ E2 ) > 0. Let Nρ j ,E j , j = 1, 2, denote N represented by ρ j , E j . If A is any closed subset of E1 \ E2 with m(A) > 0 and g ∈ b \ A), the definition of Nρ ,E says that gNρ ,E ⊂ Nρ ,E . However Nρ ,E = Nρ ,E H ∞ (C 2 2 2 2 2 2 1 1 2 2 b \ A)-invariant. The previous corollary says that m(A ∩ E1 ) = 0 and so Nρ1 ,E1 is H ∞ (C b \ γ )N ⊂ N if which is a contradiction. This says that m(E1 ∆E2 ) = 0. Notice that H ∞ (C and only if m(E) = 0. b \ γ )-invariant. Then m(E) > 0. Pick any non-zero function Suppose N is not H ∞ (C f ∈ N and notice that ρ1 = ρ2 = fi / fe almost everywhere on E.

8 The general theorem here is, for a compact subset F ⊂ C, that H ∞ (C b \ F) contains non-constant functions if and only if the analytic capacity of F is positive. However, when F ⊂ T, the analytic capacity is positive if and only if the Lebesgue measure is F is positive.

Chapter 4

Nearly invariant and the backward shift 4.1 The backward shift and pseudocontinuations For a D-inner function ϑ , form the subspace Kzϑ := H 2 (D) ∩ (zϑ H 2 (D))⊥ .

Since zϑ H 2 (D) is an S-invariant subspace of H 2 (D), then Kzϑ will be an S∗ -invariant subspace of H 2 (D), where f − f (0) S∗ f = z is the backward shift operator. It is also easy to see that Kzϑ contains the constants. In fact, by Beurling’s theorem, every S∗ -invariant subspace, which also contains the constants, takes the form Kzϑ for some D-inner function ϑ . It is well-known [16, 27] that functions in Kzϑ have special ‘continuation’ properties. Indeed, recall from (3.3.2) that for h ∈ L1 (m) (Ch)(λ ) :=

Z

T

h(ζ ) dm(ζ ) ζ −λ

denotes the Cauchy transform of h. It is known [16, p. 87] that for any f ∈ Kzϑ the meromorphic function C( f ζ ϑ )(λ ) fe(λ ) := (4.1.1) C(ζ ϑ )(λ ) on De is a pseudocontinuation of f in that the non-tangential limits of f (from D) and fe (from De ) are equal almost everywhere on T. Using the Cauchy integral formula and power series, one can prove the identity fe(λ ) =

∞ 1 1 d f ζ ϑ (−n), ∑ ∗ n−1 ϑ (λ ) n=1 λ

Chapter 4. Nearly invariant and the backward shift

48

whereb·(k) denotes the k-th Fourier coefficient and 1 ϑ (λ ) := ϑ , λ

λ ∈ De .

∗

(4.1.2)

This says that 1 fe ∈ ∗ H 2 (De ) ∀ f ∈ Kzϑ . ϑ

(4.1.3)

4.2 A new description of nearly invariant subspaces The main theorem of this section (Theorem 4.2.5 below) gives an alternate description of b \ γ ) in terms of these Kzϑ spaces. Before getting the nearly invariant subspaces of H 2 (C to this description, we make a remark about norming points.

Remark 4.2.1. If one carefully works through the proof of Theorem 3.1.2 and all the preliminary results that lead up to it, one can see that the result does not depend on the b \ γ )1 . Up to now, we have been operating under the assumption norming point for H 2 (C b \ γ ) was φγ (0), where φγ = α ◦ φG (see the appendix for that the norming point for H 2 (C b \ γ . Let us the exact formulas for α and φG ) is a certain conformal map from D onto C 2 b \ γ ) to be the origin. This yields an equivalent now change the norming point for H (C 2 b norm on H (C \ γ ) and has the added benefit that h f , 1iH 2 (C\ b γ ,0)

=

=

h f ◦ φγ , 1iH 2 (D,φ −1 (0)) Z

T

=

Z

T

=

γ

( f ◦ φγ )(ζ )d ωD,φ −1 (0) γ

( f ◦ φγ )(ζ )

f (0)

1 − |φγ−1 (0)|2

|ζ − φγ−1 (0)|2

dm(ζ )

(from (2.2.1))

and h1, 1iH 2 (C\ b γ ,0)

= h1, 1iH 2 (D,φ −1 (0)) γ

=

Z 1 − |φ −1 (0)|2 γ T

= 1.

|ζ − φγ−1 (0)|2

dm(ζ )

b \ γ , 0). This Thus the constant function 1 is a normalized reproducing kernel for H 2 (C assumption that the norming point is the origin will be especially important in the proof of Corollary 4.2.20 below. 1

When we need to emphasize the norming point z0 for H 2 (Ω) we will use the notation H 2 (Ω,z0 )

4.2. A new description of nearly invariant subspaces

49

b e g Consider the space K zϑ of meromorphic functions f on C\T by f i ∈ Kzϑ and f e = f i . 2 b \ γ ), Recall the definition of fe from (4.1.1). If N is a nearly invariant subspace of H (C ϕ is the extremal function for N, and ϑe is the De -inner factor of ϕe , we let ϑe∗ be the D-inner function 1 ∗ , z ∈ D. (4.2.2) ϑe (z) := ϑe z b From here, we form the space ϕ Kg zϑe∗ of analytic functions C \ γ .

b Remark 4.2.3. Technically speaking, Kg zϑe∗ is a space of analytic functions on C \ T and b not C \ γ . However, since ϕ has an analytic continuation across T \ γ , then so does ϑe (the De -inner part of ϕ ) [35, p. 78]. A version of Morera’s theorem [16, p. 84] says that e for each f ∈ Kg zϑe∗ , the functions f i and f i are analytic continuations of each other across g T \ γ . Thus Kg zϑe∗ , and hence ϕ Kzϑe∗ can be considered to be a space of analytic functions b \ γ. on C We will show that, in a sense, these spaces form the building blocks for every nearly invariant subspace. To explain exactly what we mean here, let N be a nearly invariant subspace with b \ γ -inner divisor equal to one and let E, ρ , F be the parameters in greatest common C Theorem 3.1.2 and let ϕ be the extremal function for N. Note, since ρ := ϕi /ϕe (3.5.2), that N contains the nearly invariant subspace fi ϕi fi 2 b 2 a.e. on γ . N0 := f ∈ H (C \ γ ) : ∈ H (D), = (4.2.4) F fe ϕe Observe how N0 is the intersection of the two closed sets b \ γ ) : fi = ϕi a.e. on γ N and f ∈ H 2 (C fe ϕe

and so N0 is closed. Moreover, ϕ ∈ N0 . Our structure theorem here is the following.

b \ γ ) with greatest common Theorem 4.2.5. Let N be a nearly invariant subspace of H 2 (C b \ γ -inner divisor equal to one. Let ϕ be the extremal function for N and let ϑe be the C De -inner factor of ϕe . Then we have the following: 1. The space N0 defined in (4.2.4) satisfies

N0 = ϕ Kg zϑe∗ .

2. For any sequence (An )n>1 of closed subsets of γ \ E with positive measure such that m(An ) → m(γ \ E) we have N=

∞ _

n=1

b \ An )N0 . H ∞ (C


50

One of the keys to proving Theorem 4.2.5 will be this following special case. b \ γ ) with greatest comProposition 4.2.6. Let N be a nearly invariant subspace of H 2 (C b \ γ -inner factor equal to one and with E(N) = γ . If ϕ is the extremal function for mon C N and ϑe is the De -inner factor of ϕ |De , then N = ϕ Kg zϑe∗ .

Proof. Recall from Corollary 3.3.1 the isometry J : N → H 2 (D) ⊕ L2 (γ , ω ) defined by fi fi Jf = , fe − ϕe ϕi ϕi

and note that since we are assuming that E(N) = γ , then JN =

1 N|D ⊕ {0}. ϕi

We also know from Corollary 3.3.1 that the first component of JN is an S∗ -invariant subspace of H 2 (D) which, since ϕ ∈ N, contains the constants. From our discussion before, we know that 1 N|D = Kzϑ ϕi for some D-inner function ϑ . This says that g N = ϕK zϑ .

To finish the proof, we will show that ϑ = cϑe∗ for some unimodular constant c. From the Wold decomposition of (JN)⊥ , in particular Proposition 3.4.2 and (3.4.9), we see that ! (JN)⊥ ∩ (H 2 (D) ⊕ {0}) =

∞ _

n=0

zn a ⊕ {0}

for some a ∈ H ∞ (D) for which, by Lemma 3.4.10, the function z 7→

a(z) ϑe∗

belongs to N0+ (D). This means that a ∈ ϑe∗ zH 2 (D) and so (JN)⊥ ∩ (H 2 (D) ⊕ {0}) ⊂ ϑe∗ zH 2 (D) ⊕ {0}.

But since the first component of JN is equal to Kzϑ = H 2 (D) ⊖ zϑ H 2 (D), we observe that zϑ H 2 (D) ⊂ ϑe∗ zH 2 (D) and thus ϑe∗ divides ϑ . On the other hand, for any f ∈ N, we know that fi /ϕi ∈ Kzϑ and so, by our previous discussion in (4.1.3), fi /ϕi has a pseudocontinuation to the function g/ϑ ∗ for some g ∈ H 2 (De ) (depending on f ). We are also assuming that E(N) = γ and so fe −

fi ϕe = 0 a.e. on γ ϕi


51

and hence (since the above identity also holds on T \ γ ) fi /ϕi has a pseudocontinuation to fe /ϕe . Since pseudocontinuations are unique (Privalov’s uniqueness theorem - [16, p. 13]) g/ϑ ∗ = fe /ϕe and so g fe = ∗ ϕe . ϑ It now must be the case that ϑ ∗ divides ϑe otherwise the restrictions to De of the functions g in K zϑ would have a common De -inner factor - which is impossible since the greatest b \ γ -inner divisor of N is one (see Lemma 3.4.12). Thus ϑ divides ϑ ∗ and thus common C e ∗ ϑ = cϑe for some unimodular constant c. Thus we see that Theorem 4.2.5 works in some special cases. In order to prove the result for a general nearly invariant subspace N, we need to take care of some technical details. Lemma 4.2.7. The Cauchy transform (C µ )(z) :=

Z

1 d µ (ζ ), ζ −z

b \ γ, z∈C

b \ γ ). of a finite complex Borel measure on γ belongs to N + (C

Proof. Consider the arcs γn , n ∈ N, with the same midpoint as γ but whose lengths satisfy ℓ(γn ) = ℓ(γ ) + 1/n. For each n ∈ N define [ 1 b\ Gn := C rγn : r > 0, |r − 1| 6 n (see Figure 4.1). On the arc (1 − 1n )γn note that

ωGn ,0 6 ωC\(1− 1 )γ ,0 b n

(4.2.8)

ωGn ,0 6 ωC\(1+ 1 )γ ,0 b n

(4.2.9)

n

(see [24, p. 307] or [59, p. 102]). In a similar way,

n

on the arc (1 + 1n )γn . Manipulations with conformal mappings (see (2.3.13)) will show that

ωC\(1− ≍ 1 b )γn ,0

1 ds, 1/2 |z − b− |1/2 |z − a− | n n

ωC\(1+ 1 )γ ,0 ≍ b n

1 ds, 1/2 |z − b+ |1/2 |z − a+ | n n

n

(4.2.10)

1 − where a− n and bn are the endpoints of (1 − n )γn . In a similar way,

n

1 + where a+ n and bn are the endpoints of (1 + n )γn (see Figure 4.1).

(4.2.11)


52

(1 + n1 )γn

γn

Gn

(1 − 1n )γn

Gn

11 00 00+ 11 an

11 00 00 11

− 11 an 00

an bn

b− n 11 00 00 11

11 00 00 11

00 11

00 + b11 n

Figure 4.1: The region Gn . If an and bn are the endpoints of γn , we note that on the rays 1 1 Ln := ran : |r − 1| 6 , Rn := rbn : |r − 1| 6 n n − + (the line segment connecting a+ n and an - respectively the line segment connecting bn − and bn ) we have the estimate

dist(z, γ ) > dist(z, γn ) > C min{|z − an|, |z − bn|}, with C > 0 independent of n. With these estimates in place, consider the functions fn on Gn defined by + − + fn (z) := (z − an )(z − a− n )(z − an )(z − bn )(z − bn )(z − bn )g(z),

where 1 g(z) := 6 z

(C µ )(k) (0)zk (C µ )(z) − ∑ k! k=0 5

!

.

For each n ∈ N, fn is bounded on Gn and thus fn ∈ H 1/2 (Gn ). Furthermore, Z

∂ Gn

| fn |1/2 d ωGn ,0 =

Z

(1− n1 )γn

+

Z

(1+ n1 )γn

+

Z

Ln

+

Z

Rn

= I + II + III + IV.

(4.2.12)


53

Observe from (4.2.8) and (4.2.10) that I6C

Z

1/2

(1− n1 )T

|g(z)|1/2 ds 6 CkgkH 1/2 (D) .

In a similar way, from (4.2.9) and (4.2.11), we have 1/2

II 6 CkgkH 1/2 (D ) . e

From (4.2.12) the integrals III and IV are uniformly bounded in n. Putting this all together, we have sup k fn kH 1/2 (Gn ) < ∞. (4.2.13) n∈N

b \ γ as n → ∞ and, for fixed k, The regions Gn increase up to C

fn (z) → f (z) := (z + 1)3(z + i)3 g(z)

uniformly on G− k as n → ∞. This means there is an M > 0, independent of n and k, so that | f (z) − fn (z)| 6 M,

z ∈ G− k ,

n > k.

From here we get | f (z)|1/2 6 21/2(M 1/2 + | fn (z)|1/2 ),

z ∈ G− k ,

n > k.

Since, by (4.2.13), the least harmonic majorant of | fn |1/2 on Gn at z = 0 is uniformly bounded in n, we see from the previous equation that | f |1/2 has a harmonic majorant, and hence a least harmonic majorant, uk on Gk and sup uk (0) < ∞. k

An application of Harnack’s inequality says, for fixed z ∈ Gk0 , that sup uk (z) < ∞. k>k0

b \ γ , we see that | f |1/2 has a Since uk pointwise increases to a harmonic function u on C 1/2 b b harmonic majorant on C \ γ , i.e., f ∈ H (C \ γ ). Using the Nevanlinna theory it follows b \ γ ). that C µ ∈ N + (C

Remark 4.2.14. 1. Lemma 4.2.7 is considered a ‘folklore’ result. With a different proof b \ γ ) for all 0 < p < 1/2. See and a little more effort, one can show that C µ ∈ H p (C [50, 75] for related results. In fact, a proof of Lemma 4.2.7 can be fashioned from [50]. We thank Dima Khavinson for pointing all this out to us.


54

2. If µ is a finite measure on [0, 1], one can adjust the proof of the previous corollary to b \ [0, 1]). In fact if ν is a finite measure on ∂ G, where show that C µ belongs to N + (C G = D \ [0, 1), we can write ν = ν |T + ν |[0, 1] and apply the fact that C(ν |T) ∈ N + (D) and the above observation to see that Cν ∈ N + (G). We will make use of this several times later on. b \ γ such that f |D ∈ H 1 (D) and f |De ∈ H 1 (De ). Corollary 4.2.15. Let f be analytic on C + b Then f ∈ N (C \ γ ). Furthermore, if, for some q > 1, Z

γ

(| fi |q + | fe |q )d ω < ∞,

b \ γ ). then f ∈ H q (C

Proof. The Cauchy integral formula says that f − f (∞) is the Cauchy transform of the finite measure dz . d µ = ( fi − fe ) 2π i Now apply Lemma 4.2.7 and Proposition 2.4.10. Here is the last technical lemma we need to prove Theorem 4.2.5.

b \ γ -inner function such that Θi is D-outer and Θe is De Lemma 4.2.16. Let Θ be a C b \ γ -inner outer. If a and b are the endpoints of γ , and Θa , Θb are the atomic singular C functions with singularity at a and b then Θ = BΘta Θsb ,

b \ γ -Blaschke product - which can be equal to one - with zeros where s,t > 0 and B is a C on T \ γ .

Proof. The hypothesis that Θi and Θe are outer functions say that the zeros - if any - of the Blaschke factor B of Θ must lie on T \ γ . Since Θi is bounded and D-outer and has boundary values equal to one almost everywhere on γ , a version of the Schwarz reflection principle shows that Θi has an analytic continuation across γ . In a similar way, Θe has an b \ γ -singular inner factor of Θ has a analytic continuation across γ . This says that the C limit of modulus one when we approach any point in the interior of γ from both within D and from within De . By a known result about limits of singular inner functions [35, p. 76] b \ γ -inner factor of Θ has no mass on the interior of γ . the singular C

Proof of Theorem 4.2.5. To show statement (1) we first notice that the extremal function for N0 is ϕ (since N0 ⊂ N and ϕ is the extremal function for N) and E(N0 ) = γ . Once we b \ γ -inner factor of N0 is one, we then use Proposition show that the greatest common C 4.2.6 to obtain the result. To this end, we observe that if ϑe is the inner factor of ϕe then

ϑe∗ ϕi ∈ H 2 (D) and

ϕe ∈ H 2 (De ). ϑe


55

The functions ϑe∗ ϕi and ϕe /ϑe are analytic continuations of each other across T \ γ and b \ γ defined by so we can use Corollary 4.2.15 to see that the analytic function ϕe on C ∗ ϑe (z)ϕ (z), z ∈ D; ϕe(z) := (4.2.17) ϕ (z)/ϑe (z), z ∈ De . b \ γ ). By the definition of N0 in (4.2.4) it follows that ϕe ∈ N0 . belongs to H 2 (C b \ γ -inner divisor Θ of N0 must divide both ϕ and ϕe. Since The greatest common C + b \ γ -inner divib ϕ /Θ ∈ N (C \ γ ), then ϕi /Θi ∈ N + (D)2 and since the greatest common C sor of N is one, we know from (3.5.3) that ϕi is D-outer and so Θi is D-outer. On the other b \ γ ), then ϕee /Θe ∈ N + (De ). But then, by the definition of ϕee in hand, since ϕe/Θ ∈ N + (C (4.2.17), Θe must divide the De -outer part of ϕe and thus Θe must indeed be De -outer. By Lemma 4.2.16 we have Θ = BΘta Θsb .

To see that B ≡ 1, notice from Lemma 4.2.16 that the zeros of B - if any - must lie in T \ γ . If there is indeed a zero z0 of B in T \ γ then ϕ must also have this zero since Θ is b \ γ -inner divisor of N0 and ϕ ∈ N0 . However, since the greatest the greatest common C b common C \ γ -inner divisor of N is one, there is an f ∈ N with f (z0 ) 6= 0. From Lemma 3.2.14, fi /ϕi ∈ H 2 (D) which is impossible due to the zero of ϕ at z0 . In a similar way, one can see that if either s or t were positive, then |ϕ | would go to zero too quickly along T \ γ for fi /ϕi to belong to H 2 (D) for every f ∈ N. This completes the proof of statement (1). To see statement (2), first note that N1 :=

_ n

b \ An)N0 H ∞ (C

is nearly invariant. Indeed, N0 is nearly invariant. Moreover, for λ 6∈ Z(N1 ) (the common b \ γ ) we have zero set of N1 ), f , g ∈ N0 with g(λ ) 6= 0, and u ∈ H ∞ (C u f − ugf (λ )g z−λ

=

f − gf (λ )g u − u(λ ) f + u(λ ) z−λ z−λ

which clearly belongs to N1 . Thus for a dense of functions h ∈ N1 we have h − hg (λ )g z−λ

∈ N1

and the nearly invariance of N1 follows. Second, we apply our main theorem (Theorem 3.1.2) to see that fi fi 2 b 2 N1 = f ∈ H (C \ γ ) : ∈ H (D), = ρ1 a.e. on E1 F1 fe b γ ), then f = g/h, where g,h ∈ H ∞ (C\ b γ ) and h is C\ b γ -outer. To show that fi ∈ N + (D) f ∈ N + (C\ ∞ b b \ γ -outer, that it suffices to show that hi is D-outer. Let N be a closure of hH (C \ γ ) and note, since h is C 2 b N = H (C \ γ ). Now follow the end of the proof of Lemma 3.4.12. 2 Indeed if

56


for some D-outer function F1 , some measurable ρ1 on γ , and some measurable set E1 ⊂ γ . The space N is described by the parameters F, ρ , E in a similar way. Since ϕ ∈ N1 and since, from (3.5.2), ρ1 and ρ are formed in the same way from ϕ , we see that ρ = ρ1 . Using (3.5.1) we see, in the same way, that F1 can be taken to be equal to F. Since N1 ⊂ N (Corollary 3.6.2) we see that E1 ⊃ E. Indeed, N1 contains a dense set of functions f for which fi / fe = ρ almost everywhere on E. To see that E1 = E almost everywhere, we proceed as follows: Suppose that m(E1 \ E) > 0, then, by the definition of the An ’s, b \ An ) be the function from (3.6.1), i.e., m(An ∩ (E1 \ E)) > 0 for some n. Let u ∈ H ∞ (C ui /ue 6= 1 on some compact subset of An of positive measure. Then f := uϕ ∈ N1 but fi ue ϕi ue = = ρ fe ui ϕe ui

and this last function is not equal to ρ almost everywhere on E1 , a contradiction.

In certain special cases, we have a refinement of Theorem 4.2.5. b \ γ ) with greatest comCorollary 4.2.18. Let N be a nearly invariant subspace of H 2 (C b \ γ -inner divisor equal to one. If ϕe is De -outer, then ϕ is C b \ γ -outer and furthermon C more, for any sequence (An )n>1 of closed subsets of γ \ E with positive measure such that m(An ) → m(γ \ E) we have N=

∞ _

n=1

b \ An )ϕ . H ∞ (C

(4.2.19)

Proof. In this case ϑe ≡ 1 and so Kg zϑe∗ = C. The identity in (4.2.19) now follows from b \ γ -inner factor of N is one, (4.2.19) shows Theorem 4.2.5. Since the greatest common C b that ϕ is C \ γ -outer. b \ γ ) which contains the Corollary 4.2.20. Let N be a nearly invariant subspace of H 2 (C constants and let F, ρ , and E be the parameters in Theorem 3.1.2.

1. The outer function F can be chosen to be equal to one and the function ρ is equal to one almost everywhere, i.e., fi 2 b N = f ∈ H (C \ γ ) : = 1 a.e. on E . (4.2.21) fe

2. For any sequence (An )n>1 of closed subsets of γ \ E with positive measure such that m(An ) → m(γ \ E), we have N=

∞ _

n=1

3. If E is open in γ and E c = γ \ E, then

b \ An ). H ∞ (C

∞ b c N = closH 2 (C\ b γ ) H (C \ E )

(4.2.22)

(4.2.23)

4.2. A new description of nearly invariant subspaces and this set is equal to n o b \ γ ) : f extends analytically across E . f ∈ H 2 (C

57

(4.2.24)

Proof. Consider the description of N via Theorem 3.1.2 with the parameters ρ , E, F. Because N contains the constants and thus ϕ ≡ 1 is the extremal function for N (see Remark 4.2.1), we see from (3.5.2) that ρ = ϕi /ϕe = 1. From (3.5.1), F can be taken to be the D-outer function such that |F|2 =

1 , 1 + w χγ

a.e.

From here one can see that b \ γ ) : fi ∈ H 2 (D) = H 2 (C b \ γ) f ∈ H 2 (C F

and so we can take F ≡ 1. This proves statement (1). Statement (2) follows directly from Corollary 4.2.18 since ϕ ≡ 1. To prove statement (3) we see that (4.2.23) follows from statement (4.2.22) with An = E c for all n. From (4.2.22) and Morera’s theorem (see [35, p. 95] or Proposition 6.2.8 below), n o b \ γ ) : f extends analytically across E . N ⊂ f ∈ H 2 (C The reverse inclusion follows from (4.2.21).

Chapter 5

Nearly invariant and de Branges spaces 5.1 de Branges spaces b \ γ ) in terms It turns out that we can also describe the nearly invariant subspaces of H 2 (C b of a de Branges-type space on C\ T. First let us review the well-known de Branges spaces on C \ R. We follow [26, p. 9 - 12]. Let Ψ be an analytic function on the upper half plane C+ = {ℑz > 0} such that ℜΨ > 0. The classical Herglotz theorem [26, p. 7] says that there is a non-negative measure µ on R and a non-negative number p such that ℜΨ(x + iy) = py +

1 π

Z ∞

y d µ (t), 2 + y2 (t − x) −∞

x + iy ∈ C+ .

(5.1.1)

The reader will recognize the above integral as the Poisson integral of µ . Extend Ψ to the lower half plane so that Ψ(z) = −Ψ(z),

z = x + iy,

y < 0.

A theorem of de Branges [26, p. 9] says that there exists a unique Hilbert space L(Ψ) of analytic functions on C \ R such that for each fixed w ∈ C \ R, the function z 7→

Ψ(z) + Ψ(w) π i(w − z)

(5.1.2)

belongs to L(Ψ) and F(w) =

*

Ψ(z) + Ψ(w) F(z), π i(w − z)

+

L(Ψ)

∀F ∈ L(Ψ).

(5.1.3)

Chapter 5. Nearly invariant and de Branges spaces

60

The previous identity says that the functions in (5.1.2) are the reproducing kernel functions for L(Ψ). Furthermore, if µ is the measure from (5.1.1), the linear transformation f 7→

1 πi

Z ∞ f (t)

t −z

−∞

d µ (t)

(5.1.4)

maps L2 (µ ) isometrically into L(Ψ) and the orthogonal complement of the range of this transformation contains only constant functions. For example, if p = 0 in (5.1.1), this map is onto. b \ T. For this, let Φ Let us create a de Branges-type space of analytic functions on C b be analytic on D with non-negative real part and extend Φ to C \ T by Φ(z) = −Φ(1/z),

z ∈ De .

By the change of variable 1+z (5.1.5) 1−z (which maps D onto C+ ) in (5.1.2) and (5.1.3), we create a unique reproducing kernel e b \ T with kernel function Hilbert space L(Φ) of analytic functions on C z 7→ i

kΦ (w, z) = (1 − w)(1 − z)

Φ(z) + Φ(w) , 2π (1 − wz)

b \ T. z, w ∈ C

That is to say, hF(·), kΦ (w, ·)iL(Φ) = F(w), e

e F ∈ L(Φ),

(5.1.6)

b \ T. w∈C

Applying the change of variable in (5.1.5) along with the integral change of variable t 7→ −i

1+ζ 1−ζ

to the integral in (5.1.4), we create an operator e e ) → L(Φ), V : L2 (µ

(V f )(z) :=

1 (1 − z) 2π

Z

T

(1 − ζ ) f (ζ ) 1 − ζz

e (ζ ). dµ

e is the pullback measure on T formed from µ (on R) via the above change of Here µ variable. This operator V is an isometry and the orthogonal complement of the range of V contains only constant functions.

5.2 de Branges spaces and nearly invariant subspaces b \ γ ) with one of these de We will now associate each nearly invariant subspace of H 2 (C e Branges-type space L(Φ). All of our results on nearly invariant subspaces so far do not

5.2. de Branges spaces and nearly invariant subspaces

61

depend on the fact that γ = {eit : −π /2 6 t 6 π } and still hold when γ is any proper sub-arc of T. We will also assume, without loss of generality, that 1 6∈ γ . b \ γ ), we let, as in (3.2.2) and (3.2.3), ϕ For our nearly invariant subspace N of H 2 (C and ψ denote the normalized reproducing kernel functions for N at z = 0 and z = ∞. From Corollary 3.2.9 (statement (2)) we know that |ϕ (1)| = |ψ (1)|. Multiplying by an appropriate unimodular constant, which will not change the fact that ϕ and ψ satisfy (3.2.1) and (3.2.3), we can assume that

ϕ (1) = ψ (1). A computation using statements (1) and (2) of Corollary 3.2.9 along with the fact that z 7→

1+z 1−z

maps D onto {ℜz > 0} will show that Φ :=

1 + zϕψ

(5.2.1)

1 − zϕψ

b \ T satisfying is an analytic function on C

ℜΦ > 0 and Φ(z) = −Φ(1/z).

e Thus from above, we can form the de Branges-type space L(Φ) along with the associated reproducing kernel kΦ in (5.1.6). Recall from (3.2.7) that the reproducing kernel for N is kλN (z) =

ϕ (λ )ϕ (z) − λ zψ (λ )ψ (z) , 1 − λz

z 6= 1/λ .

This next lemma relates the kernels kλN and kΦ . b \ γ ) with reproducing kernel Lemma 5.2.2. If N is a nearly invariant subspace of H 2 (C N kλ and Φ is given by (5.2.1), then 1 kλN (z) = |ϕ (1)|2 k1N (λ )k1N (z)kΦ (λ , z). 2 Proof. From the definition of Φ from (5.2.1) we get Φ − 1 zψ = . Φ+1 ϕ

Chapter 5. Nearly invariant and de Branges spaces

62 This means that kλN (z)

=

ϕ (λ )ϕ (z)

1 − Φ(λ )−1 Φ(z)−1 Φ(z)+1 Φ(λ )+1

1 − λz 2ϕ (λ )ϕ (z) Φ(z) + Φ(λ )

=

(Φ(λ ) + 1)(Φ(z) + 1)

1 − λz

.

Using (5.2.1) we observe that

Using the definition of k1N

ϕ ϕ − zψ = . Φ+1 2 and our assumption that ϕ (1) = ψ (1) we see that ϕ − zψ = (1 − z)ϕ (1)k1N .

Now combine these last two identities with the above computation for kλN to get kλN (z)

= =

1 Φ(z) + Φ(λ ) |ϕ (1)|2 k1N (λ )k1N (z)(1 − z)(1 − λ ) 2 1 − λz 1 2 N N Φ |ϕ (1)| k1 (λ )k1 (z)k (λ , z). 2

e Our main result relating N with L(Φ) is the following.

Theorem 5.2.3. With the assumptions above we have e N = k1N L(Φ).

Moreover, the operator

ϕ (1) f 7→ √ k1N f 2

e is an isometry from L(Φ) onto N. Proof. Given

f = ∑ c j kλNj , j

a finite linear combination of reproducing kernel functions for N (which are dense in N), define cj T f := ∑ √ ϕ (1)k1N (λ j )kλΦj 2 j and observe that hT f , T f iL(Φ) e

=

∑ j,l

=

c j cl |ϕ (1)|2 k1N (λ j )k1N (λl )kΦ (λ j , λl ) 2

∑ c j cl kλNj (λl ) j,l

=

h f , f iH 2 (C\ b γ).

(by Lemma 5.2.2)

5.2. de Branges spaces and nearly invariant subspaces

63

e By standard arguments, we can extend T to a unitary operator from N onto L(Φ). Finally, e b \ T, we have for F ∈ L(Φ) and λ ∈ C hT ∗ F, kλN iH 2 (C\ b γ)

=

= =

This says that

and is an isometry.

hF, T kλN iL(Φ) e 1 Φ N F, √ ϕ (1)k1 (λ )k (λ , ·) e 2 L(Φ)

ϕ (1) N √ k1 (λ )F(λ ). 2

ϕ (1) T ∗ F = √ k1 F 2

Chapter 6

Invariant subspaces of the slit disk 6.1 First description of the invariant subspaces In this section we use our main theorem about nearly invariant subspaces (Theorem 3.1.2) b γ from (2.3.8) to give a full description of the invariant and the conformal map α : G → C\ subspaces (under S f = z f ) of H 2 (G). Let us get started with a few preliminary observations. Proposition 6.1.1. A subspace M ⊂ H 2 (G) is invariant if and only if M is H ∞ (D)invariant, i.e., gM ⊂ M for every g ∈ H ∞ (D).

Proof. One direction of the argument is obvious. For the other, suppose that f ∈ M and φ ∈ H ∞ (D). Let (φn )n>1 be a sequence of analytic polynomials such that φn → φ weak-∗ in H ∞ (D)1 , i.e., φn → φ pointwise in D and the sup-norms of φn are uniformly bounded in n. Since φn f → φ f pointwise in G and the H 2 (G)-norms of φn f are uniformly bounded, it follows that φn f → φ f weakly in H 2 (G) [14, p. 272]. Note that φn f ∈ M (since M is invariant) and φ f belongs to the weak-closure of M. By standard functional analysis, the weak-closure of M is equal to its norm closure [20, p. 129] and so φ f ∈ M. Proposition 6.1.2. A non-zero subspace M ⊂ H 2 (G) is H ∞ (G)-invariant if and only if M = ΘH 2 (G) for some G-inner function Θ.

Proof. Recall from (2.1.3) that if φG is a conformal map from D onto G, the composition operator CφG f = f ◦ φG

is a unitary operator from H 2 (G) onto H 2 (D). Now on to the proof. One direction is clear. For the other, suppose M 6= {0} and H ∞ (G)-invariant. Then CφG M is a non-zero 1

The sequence of Cesàro polynomials of φ will work [46, p. 19].

Chapter 6. Invariant subspaces of the slit disk

66

H ∞ (D)-invariant subspace of H 2 (D). By Beurling’s theorem classical theorem which characterizes the invariant subspaces of H 2 (D) [35, p. 82], CφG M = IH 2 (D), where I is a D-inner function. Finally, notice that M = (I ◦ φG−1 )H 2 (G) and, by definition, I ◦ φG−1 is G-inner. Corollary 6.1.3. An invariant subspace M ⊂ H 2 (G) is nearly invariant.

Proof. By Proposition 3.1.1, it suffices to show that whenever f , g ∈ M and h ∈ M⊥ , * f − f (a)g + g , h = 0 ∀a ∈ G \ (Z(M) ∪ g−1({0})). z−a

(6.1.4)

Recall that Z(M) is the set of common zeros of M. Let W (a) be equal to the meromorphic function on G defined by the left-hand side of the above equation. Notice that W can be written in the form W (a) =

Z

d µ1 (ζ ) + T ζ −a

Z

d µ2 (x) f + (a) g [0,1] x − a

Z

d µ3 (ζ ) f + (a) g T ζ −a

Z

d µ4 (x) . [0,1] x − a

One can argue, using some ideas from Remark 4.2.14, that W is in the Nevanlinna class of G and hence has finite non-tangential limits almost everywhere on ∂ G. Since M is invariant, we have * f − f (a)g + g , h = 0 ∀|b| > 1. z−b Thus * f − f (a)g + * f − f (a)g + g g W (a) = ,h − ,h , z−a z−b

a ∈ G \ g−1({0}), |b| > 1.

For r ∈ (0, 1) and ζ ∈ T, let a = rζ and b = ζ /r. Apply Fatou’s jump theorem (Theorem 3.3.3) to see that lim W (rζ ) = 0 a.e. ζ ∈ T. (6.1.5) r→1−

Observe in the inner product how the contribution from the parts of the integral on the slit [0, 1] cancel out in the limit. Using the well-know fact that if a Nevanlinna function has vanishing radial boundary values on a set of positive measure then this function must vanish identically, along with (6.1.5), we see that W ≡ 0. This proves (6.1.4). We are now ready for our first description of the invariant subspaces of H 2 (G). We will see another description of them later on in Theorem 6.2.1.

6.1. First description of the invariant subspaces

67

Corollary 6.1.6. Let M be a non-trivial invariant subspace of H 2 (G) with greatest common G-inner divisor ΘM . Then there exists a D+ -outer function F, a measurable set E ⊂ [0, 1], and a measurable function ρ : [0, 1] → C such that f |D+ 2 − 2 + M = ΘM · f ∈ H (G) : ∈ H (D+ ), f = ρ f a.e. on E . F b \ γ from (2.3.8). Recall the unitary operProof. Consider the conformal map α : G → C −1 2 2 b ator Cα −1 h = h ◦ α from H (G) onto H (C \ γ ). We will now use Proposition 3.1.1 to b \ γ (and not in the common zero set show that Cα −1 M is nearly invariant. Indeed if λ ∈ C −1 of Cα −1 M) and f ∈ M with ( f ◦ α )(λ ) = 0, we need to show that f ◦ α −1 ∈ Cα −1 M. z−λ

But this is equivalent to showing that f ∈ M. α −λ However,

f z − α −1(λ ) f = . α −λ α − λ z − α −1(λ )

The second factor belongs to M, since M is nearly invariant (Corollary 6.1.3). The first factor belongs to H ∞ (D)2 and M is H ∞ (D)-invariant (Proposition 6.1.1). Thus f /(α − λ ) ∈ M and so Cα −1 M is nearly invariant. Since M is H ∞ (D)-invariant

α ([0, 1]) = γ ′′ := {eit : 3π /2 6 t 6 2π }, b \ γ ′ )-invariant, where γ ′ := {eit : then Cα −1 M is not only nearly invariant but is also H ∞ (C 0 6 t 6 π }. By Theorem 3.1.2, fi 2 b 2 Cα −1 M = Θ · f ∈ H (C \ γ ) : ∈ H (D), fi = ρ fe a.e. on E F b \ γ -inner Θ, some D-outer F, some measurable E ⊂ γ ′′ (note that C −1 M is for some C α b \ γ ′ )-invariant), and some measurable ρ : E → C. Thus H ∞ (C e · f ∈ H 2 (G) : f |D+ ∈ H 2 (D+ ), f + = ρe f − a.e. on Ee , M=Θ Fe

e := Θ ◦ α is G-inner, Fe := F ◦ α is D+ -outer (since α (D+ ) = D), Ee := α −1 (E) ⊂ where Θ [0, 1] (since E ⊂ γ ′′ and α ([0, 1]) = γ ′′ ), and ρe = ρ ◦ α . 2

√ √ See (2.3.8) and notice how α is analytic on D \ {i(1 − 2)} with a simple pole at i(1 − 2).


68

e ρe, and Ee are (essentially) unique. Remark 6.1.7. By Corollary 3.6.3, the parameters Θ, We also have the following version of Corollary 4.2.20.

Corollary 6.1.8. Let A be a closed subset of [0, 1]. For f ∈ H 2 (G) the following are equivalent. 1. f ∈ closH 2 (G) H ∞ (D \ A); 2. f + = f − almost everywhere on [0, 1] \ A; 3. f has an analytic continuation across [0, 1) \ A.

6.2 Second description of the invariant subspaces The description of the invariant subspaces of H 2 (G) in Corollary 6.1.6 depends on the somewhat unnatural use of the D+ -outer function F. This next result is an alternate, and perhaps more natural, description. For ε ∈ (0, 1) let Gε := D \ [−ε , 1). Theorem 6.2.1. For an invariant subspace M of H 2 (G), let ΘM , E, and ρ be as in Corollary 6.1.6. Then for every ε ∈ (0, 1), there is a Gε -outer function Fε such that f M = ΘM · f ∈ H 2 (G) : ∈ H 2 (Gε ), f + = ρ f − a.e. on E . Fε

The proof of this theorem needs quite a few preliminaries. Let ω , ωε , and ω+ be harmonic measure for G (respectively Gε and D+ ) at some common point in D+ . If Ω = G (or Gε or D+ ), we have d ω ≍ wΩ

ds 2π

and wΩ ≍ |ψΩ′ |,

(6.2.2)

where ψ is a conformal map from Ω onto D and ds is arc length measure on ∂ Ω. See this from (2.3.5) for G and Gε and (2.2.3) for D+ . We will use the notation w := wG ,

wε := wGε ,

w+ := wD+ .

Remark 6.2.3. If ω is harmonic measure for G, we will assume that ψ −1 (0) ∈ D+ and

ω = ωψ −1 (0) and so by (2.3.5) ds . (6.2.4) 2π Having ω precisely as in (6.2.4) will become important in one of the technical lemmas below (see Lemma 6.2.14). d ω = |ψ ′ |

6.2. Second description of the invariant subspaces

69

Recall, from our discussion of the estimates of harmonic measure in Chapter 2, that if η is one of the corners of ∂ Ω with opening θ (0 < θ 6 2π ), then π

|ψ ′ (ξ )| ≍ |ξ − η | θ −1 ,

ξ ≈ η.

(6.2.5)

Our first technical lemma is standard [24, p. 307] [59, p. 102]. Lemma 6.2.6. For each ε ∈ (0, 1), ωε 6 ω on ∂ G. Lemma 6.2.7.

1. log w ∈ L1 (∂ G, ω ).

2. For each ε ∈ (0, 1),

Z

T+

iθ

log wε wε ds < ∞, w+ ∪[−ε ,1]

where T+ := {e : 0 6 θ 6 π }

Proof. From (6.2.2) and (6.2.5) we have w+ ≍ |ξ + 1|, w+ ≍ |ξ − 1|,

w ≍ 1 for ξ ≈ −1;

wε ≍ 1,

wε ≍ |ξ − 1|,

wε ≍ |ξ + ε |−1/2 ,

w+ ≍ 1,

w+ ≍ 1,

wε ≍ 1,

w ≍ |ξ − 1| for ξ ≈ 1; w ≍ 1 for ξ ≈ −ε ;

w ≍ |ξ |−1/2

for ξ ≈ 0.

Furthermore, if one stays away from the points ξ = 1, −1, 0, −ε , the functions w+ , wε , w are continuous and bounded away from zero. The result follows. We will also make use of the following Morera-type theorem [35, p. 95]. Recall the definition of the Hardy-Smirnov classes E 1 from (2.4.8). Proposition 6.2.8. Suppose f1 ∈ E 1 (D+ ) and f2 ∈ E 1 (D− ) with lim f1 (x + iy) = lim f2 (x + iy)

y→0+

y→0−

almost everywhere on [−1, 1]. Then the function f1 (z), z ∈ D+ ; g(z) := f2 (z), z ∈ D− has an analytic continuation to D. Proof. Using the E 1 version of the Cauchy integral formula (Proposition 2.4.12) we have g(z) =

1 2π i

I

∂ D±

g(ξ ) dξ , ξ −z

z ∈ D± .

Also notice that g(z) =

1 2π i

I

∂ D+

g(ξ ) 1 dξ + ξ −z 2π i

I

∂ D−

g(ξ ) dξ , ξ −z

z ∈ D+ ∪ D− .


70 But since

lim g(x + iy) = lim g(x + iy)

y→0+

y→0−

almost everywhere on [−1, 1], the integrals over [−1, 1] cancel out and so g(z) =

1 2π i

I

T

g(ξ ) dξ . ξ −z

The above integral defines an analytic function on D and this proves the result. This next technical lemma is due to Smirnov [72] (see also [49, p. 319]). Lemma 6.2.9. For each g ∈ N + (D) there is a sequence (gn )n>1 ⊂ H ∞ (D) such that 1. |gn (z)| 6 |g(z)| for all z ∈ D; 2. gn → g pointwise on D.

Proof. Factor g as g = θ h, where θ is D-inner and h is D-outer [35, p. 74]. For each n ∈ N let hn be the bounded D-outer function whose boundary function satisfies |hn (ζ )| =

|h(ζ )|, if |h(ζ )| 6 n; n, if |h(ζ )| > n.

for almost every ζ ∈ T. We leave it to the reader to check, using properties of outer functions [35, p.73], that the functions gn := θ hn , n ∈ N, have the desired properties. Remark 6.2.10. Though not needed for what follows, we point out that if f is analytic on D and is a pointwise limit of a sequence of bounded analytic functions with increasing moduli, then f ∈ N + (see [49, p. 319] for a proof). For an invariant subspace M of H 2 (G) and λ0 ∈ G \ Z(M), where Z(M) is the set of common zeros for M, let kλM ϕ := M0 kkλ k 0

be the normalized reproducing kernel function for M (or equivalently the ‘extremal function’ for M) at λ0 . Note that ϕ ∈ M and h f ,ϕi =

f (λ0 ) ∀ f ∈ M, ϕ

(6.2.11)

hϕ , ϕ i = 1. These next two technical lemmas point out some special properties of this extremal function. But first we pause for a few remarks.


71

b \ γ is the conformal map from (2.3.8) and N = Remark 6.2.12. 1. Suppose α : G → C Cα −1 M. From the proof of Corollary 6.1.6 we know that N is nearly invariant. If Φ :=

k0N kk0N k

is the normalized reproducing kernel function for N at 0, we can use the unitary operator Cα −1 to show that if we assume that λ0 = α −1 (0), then kλM0 = Cα k0N and consequently

ϕ = Φ ◦ α.

(6.2.13)

2. In what follows below, we need to be clear on how we represent the inner product in H 2 (G) as an integral. When Ω is Jordan domain with piecewise analytic boundary, the inner product in H 2 (Ω) can be written as h f , gi =

Z

∂Ω

f g d ωψ −1 (0),Ω =

Z

∂Ω

f g|ψ ′ |

ds , 2π

where ψ : Ω → D. For the slit domain G = D \ [0, 1) the expression Z

∂G

f gd ω ,

is not quite right since we need to take into account the fact that f + g+ and f − g− are, in general, different. Thus we will use the notation Z

∂G

f gd ω ∗

to mean Z

T

f (ζ )g(ζ )w(ζ )

|d ζ | + 2π

Z 1 0

( f + (x)g+ (x) + f − (x)g− (x))w(x)

dx , 2π

where w = |ψ ′ |. Note that we really should have w+

and w− in the above expression. However, recall from (2.3.2) that = Also observe from Proposition 2.3.4 that this last expression is precisely h f , gi, the inner product in H 2 (G). w+

w− .

Lemma 6.2.14. If the greatest common G-inner divisor of an invariant subspace M of H 2 (G) is equal to one, then the normalized reproducing kernel function ϕ for M at λ0 ∈ G \ Z(M) extends analytically across T \ {1}.

Proof. For each |λ | > 1, note that

z − λ0 ∈ H ∞ (D) z−λ


72 and so, by Proposition 6.1.1, z − λ0 f ∈M z−λ

∀ f ∈ M \ {0}.

Thus, by the reproducing property of ϕ at λ0 (see (6.2.11)), we have Z z − λ0 z − λ0 f (z)ϕ (z)d ω ∗ (z) = f , ϕ = 0. z−λ ∂G z − λ

(6.2.15)

Take note of Remark 6.2.12. Combining Fatou’s jump theorem (Theorem 3.3.3) and (6.2.15), we get lim

r→1−

Z

z − λ0 f (z)ϕ (z)d ω ∗ (z) = ζ (ζ − λ0) f (ζ )ϕ (ζ )w(ζ ) ∂ G z − rζ

a.e. ζ ∈ T.

(6.2.16)

Notice how the contribution from the integrals over the slit cancels out in the limit. From Remark 6.2.3 (in particular (6.2.4)) w(ζ ) is equal to |ψ ′ (ζ )| and from elementary facts about conformal mappings, ψ ′ is analytically continuable across T \ {1} and the analytic continuation has no√zeros in an open neighborhood of T \ {1}.3 It follows, for some appropriate branch of ·, that the function p W (z) := ψ ′ (z)

has the same property. Note that w(ζ ) = W (ζ )W (ζ ) for ζ ∈ T \ {1}. This means that the identity in (6.2.16) can be re-written as

ζ ϕ (ζ )W (ζ ) =

1 lim (ζ − λ0) f (ζ )W (ζ ) r→1−

Z

z − λ0 f (z)ϕ (z)d ω ∗ (z) ∂ G z − rζ

a.e. ζ ∈ T.

(6.2.17) Now select a ζ0 ∈ T \ {1} and an open disk ∆ := {|z − ζ0 | < r} contained in the region of analyticity of W and such that λ0 6∈ ∆. Consider the function F on ∆∩De defined by 1 1 1 F(λ ) := ϕ W . λ λ λ Observe that F is analytic on ∆ ∩ De , F|T is almost everywhere equal to the left-hand side of (6.2.17), and, as a consequence, is integrable on ∆ ∩ T. (Note that ϕ is integrable on ∆ ∩ T and W is bounded on T.) Adjusting the radius of ∆ slightly, we can assume that F is also bounded on (∂ ∆) ∩ De . But since F ∈ N + (∆ ∩ De ) and has integrable boundary values, we see from Proposition 2.4.10 that F ∈ E 1 (∆ ∩ De ). Now consider the function F1 on ∆ ∩ D defined by F1 (λ ) := 3

1 (λ − λ0 ) f (λ )W (λ )

Z

∂G

z − λ0 f (z)ϕ (z)d ω ∗ (z). z−λ

(6.2.18)

One can also see this by looking at the exact form of ψ which one can compute from the appendix.


73

Since the greatest common G-inner divisor of M is equal to one, we conclude that F1 ∈ N + (∆ ∩ D)4 and, adjusting the radius of ∆, is bounded on (∂ ∆) ∩ D. Note also from (6.2.17) and the discussion in the previous paragraph, that F1 is integrable on ∆ ∩ T. Thus F1 ∈ E 1 (∆ ∩ D). Finally, F(ζ ) = F1 (ζ ) for almost every ζ ∈ ∆ ∩ T and so, by Proposition 6.2.8 (Morera’s theorem), F1 is an analytic continuation of F across ∆ ∩ T. Since F has an analytic continuation to ∆ and W is analytic on ∆, one can look at the formula defining F to see that ϕ has an analytic continuation across ∆ ∩ T. The lemma now follows. Corollary 6.2.19. The normalized reproducing kernel function ϕ in Lemma 6.2.14 has no zeros on [−1, 0). The proof of Corollary 6.2.19 requires some information about the boundary values of the D+ -outer function F in the statement of Corollary 6.1.6. Recall from the proof of Corollary 6.1.6 that F = F1 ◦ (α |D+ ), b \ γ is from (2.3.8) and F1 is the D-outer function from the proof of where α : G → C Theorem 3.1.2 (see (3.5.1)). More precisely, let Φ be the normalized reproducing kernel function at the origin for the nearly invariant subspace Cα −1 M and let F1 be the D-outer function whose non-tangential boundary function satisfies |F1 |2 =

|Φi |2 1 + wγ χγ |Φe |2

a.e on T,

where Φi (ζ ) = lim Φ(rζ ), r→1−

Φe (ζ ) = lim Φ(rζ ) r→1+

a.e. ζ ∈ T.

Let us now compute the non-tangential boundary function for |F|. For ζ ∈ T notice that |F1 (ζ )|2 = ∡ lim z→ζ

|Φ(z)|2 , 1 + wγ (ζ )χγ (ζ )|Φ( 1z )|2

where ∡ lim denotes the non-tangential limit as z → ζ , z ∈ D. Thus for almost every ξ ∈ T 4 By (6.2.16), the non-tangential boundary values of F are the same for every f ∈ M \ {0}. By Privalov’s 1 uniqueness theorem [52, p. 62], the definition of F1 is independent of f . Using the fact that the greatest common divisor of M is one, it can be argued, by adjusting f , that no part of an G-inner factor may appear in the denominator of the definition of F1 .


74 with 0 < arg(ξ ) < π we have |F(ξ )|2

= = = = = =

|F1 ◦ α (ξ )|2

|Φ ◦ α (rξ )|2 r→1− 1 + (wγ ◦ α )(ξ )|Φ( lim

1 )|2 α (rξ )

|Φ ◦ α (rξ )|2 r→1− 1 + wG (ξ )|Φ( 1 )|2 lim

lim

r→1−

α (rξ ) |Φ ◦ α (rξ )|2

(since α (D+ ) = D)

(since wγ ◦ α = wG )

1 + wG (ξ )|Φ ◦ α (rξ )|2

(since 1/α (rξ ) = α (rξ ))

|Φ ◦ α (ξ )|2

1 + wG (ξ )|Φ ◦ α (ξ )|2 |ϕ (ξ )|2

(since ϕ = Φ ◦ α from (6.2.13)).

1 + wG (ξ )|ϕ (ξ )|2

For almost every x ∈ [0, 1] we have |F(x)|2

=

lim |F1 ◦ α (x + iy)|2

y→0+

=

|Φ ◦ α (x + iy)|2 1 y→0+ 1 + wG (x)|Φ( )|2

=

|Φ ◦ α (x + iy)|2 lim y→0+ 1 + wG (x)|Φ ◦ α (x − iy)|2

lim

α (x+iy)

=

(since 1/α (z) = α (z))

|ϕ + (x)|2 1 + wG (x)|ϕ − (x)|2

since α (x + iy) ∈ D, α (x − iy) ∈ De , α + (x) = α − (x), α ([0, 1]) = {eiθ : 32π 6 θ 6 2π }. For x ∈ (−1, 0) we have |F(x)|2 = |ϕ (x)|2 . To summarize, F is the D+ -outer function whose boundary function (almost everywhere) satisfies  |ϕ (ξ )|2   , ξ ∈ T+ ;    1 + wG(ξ )|ϕ (ξ )|2 |F(ξ )|2 = |ϕ (ξ )|2 , ξ ∈ [−1, 0];    |ϕ + (ξ )|2   , ξ ∈ [0, 1]. 1 + wG (ξ )|ϕ − (ξ )|2

In the above definitions, and for what follows, we will use the notation T+ := {eiθ : 0 6 θ 6 π } and T− := {eiθ : π 6 θ 6 2π }.

(6.2.20)


75

Proof of Corollary 6.2.19. Let λ ∈ (−1, 0) and let Iλ be a closed sub-interval of (−1, 0) that contains λ in its interior. From the formula for F in (6.2.20), notice that |F| = |ϕ | a.e. on (−1, 0).

(6.2.21)

Notice also from elementary facts about conformal maps and (6.2.2) that w+ |Iλ is bounded above and below.

(6.2.22)

Since λ ∈ G and ΘM ≡ 1, there must be an f ∈ M such that f is never zero on Iλ .

(6.2.23)

But since f |D+ /F ∈ H 2 (D+ ) we have, from applying (6.2.22) followed by (6.2.23) followed by (6.2.21), ∞>

Z Iλ

2 Z f w+ dx > c F I

2 Z Z f 1 1 dx > c dx = c dx. F 2 |F| | ϕ |2 I I λ λ λ

Since ϕ is analytic in a neighborhood of Iλ , the only way that Z

Iλ

1 dx < ∞ |ϕ |2

is for ϕ to have no zeros on Iλ . Thus we have shown that ϕ has no zeros on (−1, 0). We will now argue that ϕ (−1) 6= 0. Let J be the arc of the unit circle subtended by the points −1 and i. Let f ∈ M \ {0} and let f1 be the D-outer function whose boundary values satisfy   1 , a.e. on J; | f1 | := (6.2.24) |f|  1, a.e. on T \ J. 5

By the definition of D-outer we have Z ζ +z log | f (ζ )|dm(ζ ) f1 (z) = exp − J ζ −z

and so, since the integration is over J, f1 is bounded on [0, 1). Clearly we have f f1 ∈ N + (G)6 and f1 f ∈ L2 (∂ G, ω ). By Proposition 2.4.10, f1 f ∈ H 2 (G). Use Lemma 6.2.9 to produce a sequence (gn )n>1 in H ∞ (D) with gn → f1 pointwise in D and |gn | 6 | f1 | on D. By Proposition 6.1.1, gn f ∈ M for each n. Moreover, gn f → f1 f pointwise in G. We also see that |gn f |2 6 | f1 f |2 on G and so, by the harmonic majorant definition of the norm on H 2 (G), kgn f kH 2 (G) is uniformly bounded in n. Thus gn f → f1 f weakly and so f1 f ∈ M. 5 By (2.4.3), log | f | ∈ L1 (∂ G, ω ) and from (2.3.6) d ω ≍ d θ on J. Thus it follows that log | f | ∈ L1 (T,m) and 1 so such a D-outer function actually exists. 6 Observe that f ∈ H 2 (G) ⊂ N + (G) and f1 is D-outer and hence, by Proposition 2.4.5, f1 is G-outer.


76 Moreover, from (6.2.2) and (6.2.5),

w+ |J ≍ |z + 1|.

(6.2.25)

Since f1 f ∈ M, then f1 f |D+ /F ∈ H 2 (D+ ) and so, by using (6.2.24) followed by (6.2.20) followed by (6.2.25), Z Z Z Z f1 f 2 w+ w+ |ζ + 1| w+ |d ζ | > |d |d |d ζ |. ∞> ζ | > ζ | > c 2 2 2 F |F| | ϕ | | J J J J ϕ (ζ )| Since ϕ is analytic in a neighborhood of −1 (Lemma 6.2.14), the only way this last integral can be finite is for ϕ (−1) 6= 0.

Lemma 6.2.26. For each ε ∈ (0, 1), the D+ -outer function from (6.2.20) satisfies Z

T+ ∪[−ε ,1]

| log |F||wε ds < ∞.

Proof. Using (6.2.20), write |ψ1 |2 , 1 + wG |ψ2 |2

|F|2 = We see that Z

Z

2

T+ ∪[0,1]

| log |F| |wε ds 6

T+ ∪[0,1]

Z

6

T+ ∪[0,1]

a.e. on T+ ∪ [0, 1].

| log |F|2 |wds (by Lemma 6.2.6) | log |ψ1 |2 |wds +

Z

T+ ∪[0,1]

log(1 + w|ψ2 |2 )wds.

The first integral in the previous line converges since ψ1 is part of the boundary function for ϕ and ϕ ∈ H 2 (G) \ {0} (see (2.4.3)). For the second integral, use the inequality log(1 + y) 6 1 + | logy|,

y > 0,

to show that this integral is bounded above by Z

T+ ∪[0,1]

wds +

Z

T+ ∪[0,1]

| log w|wds +

Z

T+ ∪[0,1]

| log |ψ2 |2 |wds.

The first integral clearly converges. The second integral converges by Lemma 6.2.7(1) while the third integral converges since ψ2 is part of the boundary function for ϕ and ϕ ∈ H 2 (G). We are now left with showing that the integral Z

[−ε ,0]

| log |F||wε ds

converges. But this one is easy since, by (6.2.20), |F| = |ϕ | a.e. on [−ε , 0] and ϕ ∈ H 2 (Gε ) \ {0}. (This last fact follows from the fact that ϕ ∈ H 2 (G) and Gε ⊂ G - see (2.1.4)). This completes the proof.


77

With these technical details out of the way, we are finally ready for the proof of Theorem 6.2.1. Proof of Theorem 6.2.1. Let ε ∈ (0, 1). We leave it to the reader to use Lemma 6.2.7 and Lemma 6.2.26 to verify that there is a Gε -outer function Fε whose boundary function satisfies wε + 2 |(Fε )+ |2 = |F | a.e. on [−ε , 1]; w+ wε |Fε |2 = |F|2 a.e. on T+ ; w+ |Fε |2 = 1 a.e. on T− ;

|(Fε )− |2 = 1 a.e. on [−ε , 1].

To finish the proof, we need to show that for f ∈ H 2 (G) f |D+ f ∈ H 2 (D+ ) ⇔ ∈ H 2 (Gε ), F Fε or equivalently that I1 :=

Z

∂ D+

if and only if Z

Z f 2 I2 := wε ds + T Fε [−ε ,1]

By the construction of Fε above we have 6 I1 +

I2

6

Z

2 f w+ ds < ∞ F

+ 2 − 2 ! f f (Fε )+ + (Fε )− wε ds < ∞.

| f − |2 wε ds +

[−ε ,1] I1 + ck f k2H 2 (G)

Z

T−

| f |2 wε ds

(by Lemma 6.2.6)

and, since |F|2 = |ϕ |2 on [−1, −ε ] and ϕ is non-zero on [−1, −ε ] (Corollary 6.2.19), I1

Z

| f |2 w ds 2 + [−1,−ε ] |F|

6

I2 +

6

I2 + ck f k2H 2 (D+ )

6

I2 + ck f k2H 2 (G)

(by (2.1.4)).

Remark 6.2.27. As one can see from the very end of the proof of Theorem 6.2.1, the fact that ε > 0 is important since the constant c in the last two lines of the proof depends on ε . We do not know whether or not the condition ‘ f /Fε ∈ H 2 (Gε )’ (where Fε is some Gε outer function) can be replaced by ‘ f /F ∈ H 2 (G)’ (where F is some G-outer function).

Chapter 7

Cyclic invariant subspaces 7.1 Two-cyclic subspaces If M is an invariant subspace of H 2 (G), the proof of Corollary 6.1.6 shows that N := b \ γ ). We know from Corollary 3.2.9 that Cα −1 ◦ M is a nearly invariant subspace of H 2 (C if {0, ∞} is not a subset of the common zeros of N and Φ and Ψ are the normalized reproducing kernels at z = 0 and z = ∞, then the smallest nearly invariant subspace containing Φ and Ψ is equal to N. From Remark 6.2.12 we also see that Φ ◦ α is the normalized reproducing kernel for M at α −1 (0) while Ψ ◦ α is the normalized reproducing kernel at α −1 (∞). Theorem 7.1.1. If M is a non-trivial invariant subspace of H 2 (G), then M=

_

{zn (Φ ◦ α ), zm (Ψ ◦ α ) : n, m ∈ N0 } .

Proof. Without loss of generality, we assume that α −1 (0) and α −1 (∞) do not belong to the common zero set of M. We know that Cα −1 M = NΦ,Ψ , where NΦ,Ψ is the smallest nearly invariant subspace containing Φ and Ψ. Thus we have MΦ◦α ,Ψ◦α ⊂ M = Cα NΦ,Ψ , where MΦ◦α ,Ψ◦α is the smallest invariant subspace of H 2 (G) containing Φ ◦ α and Ψ ◦ α . b γ ) which contains This means that Cα −1 MΦ◦α ,Ψ◦α is a nearly invariant subspace of H 2 (C\ Φ and Ψ and so, by definition, Cα −1 MΦ◦α ,Ψ◦α = NΦ,Ψ . The result now follows. For general functions f , g ∈ H 2 (G), when is the invariant subspace generated by f and g equal to all of H 2 (G)? Theorem 7.1.2. If f , g ∈ H 2 (G) \ {0}, then _

{zn f , zm g : n, m ∈ N0 } = H 2 (G)

Chapter 7. Cyclic invariant subspaces

80

if and only if f and g have no non-trivial common G-inner factor and the set f + (x) g+ (x) x ∈ [0, 1) : − = − f (x) g (x) has Lebesgue measure zero. Proof. One direction is easy. For the other direction, let M be the invariant subspace generated by f and g. By Corollary 6.1.6, h|D+ − 2 + 2 ∈ H (D+ ), h = ρ h a.e. on E . M = Θ · h ∈ H (G) : F Since f and g have no common G-inner factor, it must be the case that Θ ≡ 1. Also, since the functions f + / f − and g+ /g− are equal almost nowhere, E has measure zero. Thus f |D+ M = h ∈ H 2 (G) : ∈ H 2 (D+ ) . F

This means that M is H ∞ (G)-invariant and so, by Proposition 6.1.2, M = Θ1 H 2 (G) for some G-inner function Θ1 . But again, since f and g have no common G-inner factor, we must have Θ1 ≡ 1 and so M = H 2 (G). √ 2 Example 7.1.3. The invariant subspace generated by the functions 1 and z is H (G).

7.2 Cyclic subspaces Theorem 7.1.1 says that every invariant subspace M is 2-cyclic in the sense that it is generated by two functions. Do we really need both functions to generate M? Is M always cyclic, i.e., is there a single f ∈ M so that M = [ f ] :=

_

{zn f : n ∈ N0 }?

When M = H 2 (G), results form [4]1 show that M is not cyclic (see also Remark 7.2.1 below). What are the cyclic invariant subspaces of H 2 (G)? Remark 7.2.1. It is easy to see that [ f ] ⊂ M(ρ ) := h ∈ H 2 (G) : h+ = ρ h− a.e. on [0, 1] ,

where ρ = f + / f − almost everywhere. Suppose that an invariant subspace M 6= {0}, with parameters Θ, Fε (0 < ε < 1), ρ , E from Theorem 6.2.1 (note that Θ, ρ , and E are essentially unique - Corollary 3.6.3) is cyclic. Then m1 ([0, 1] \ E) = 0. Here m1 is Lebesgue measure on [0, 1]. Indeed, suppose m1 ([0, 1] \ E) > 0. Then there is a closed subset F of [0, 1] \ E with m1 (F) > 0. Using (3.6.1), one produces a g ∈ H ∞ (D \ F) \ {0} with g+ 6= 1 g−

1

The follow-up papers [2, 3] discuss other cyclicity problems.

7.2. Cyclic subspaces

81

on some compact subset of F of positive measure. From the definition of M we see that gM ⊂ M. Thus if [ f ] = M, then g f ∈ M. However M = [ f ] ⊂ M(ρ ) as above and so (g f )+ /(g f )− = ρ almost everywhere on [0, 1]. But g was constructed so that this last equality can not hold almost everywhere on F. Thus M is cyclic ⇒ m1 ([0, 1] \ E) = 0. We will see in Example 8.2.13, using an analysis of the essential spectrum of S|M, that the other direction does not hold. The next few results compute [ f ] for certain reasonably well-behaved f ∈ H 2 (G).

Theorem 7.2.2. Suppose both h and 1/h belong to H 2 (G). Then [h] = M(ρ ), where ρ = h+ /h− . Proof. So far we have [h] ⊂ M(ρ ). To see the other direction, suppose f ∈ M(ρ ). Using the fact that 1/h ∈ H 2 (G) and the Cauchy-Schwarz inequality, we see that g := f /h ∈ H 1 (G) and so by (2.1.4), g|D+ ∈ H 1 (D+ )

and g|D− ∈ H 1 (D− ).

(7.2.3)

Let q := w3 ◦ w2 ◦ w1 be a conformal map from D onto D+ (see the appendix). A computation shows that the function (1 + q)(1 − q)q′ is bounded. From the conformal invariance of the Hardy spaces and (7.2.3), we have that g|D+ ◦ q ∈ H 1 (D) and consequently, the function g1 (z) := (1 + z)(1 − z)g(z) has the property that g1 |D+ ∈ E 1 (D+ ), i.e., (g1 |D+ ◦ q)q′ ∈ H 1 (D) - see (2.4.9). In a similar way, g1 |D− ∈ E 1 (D− ). Using the Cauchy integral formula (Proposition 2.4.12) we have g1 (z) =

1 2π i

I

∂ D±

g1 (ζ ) dζ , ζ −z

z ∈ D± .

However, (g1 )+ = (g1 )− almost everywhere on [−1, 1] and so for all z ∈ D \ [−1, 1] we have I I I 1 1 1 g1 (ζ ) g1 (ζ ) g1 (ζ ) g1 (z) = dζ + dζ = dζ . 2π i ∂ D+ ζ − z 2π i ∂ D− ζ − z 2π i T ζ − z

Notice in the above calculation how the integrals on [−1, 1] cancel each other out. This means that g1 is a Cauchy transform of a measure on T and consequently g1 has an analytic continuation across [−1, 1] to a function which belongs to H p (D) for all 0 < p < 1 [32, p. 39]. In particular, g ∈ N + (D). By Lemma 6.2.9, there is a sequence (gn )n>1 in H ∞ (D) such that gn → g pointwise in D as n → ∞ and |gn | 6 |g| on D. Hence gn h → f pointwise in G and |gn h|2 6 |gh|2 = | f |2 on G. This last inequality says that the H 2 (G) norms of gn h are uniformly bounded. Thus gn h ∈ [h] (Proposition 6.1.1) and gn h → f weakly in H 2 (G) which means that f ∈ [h]. It is routine to show that if f and 1/ f belong to H 2 (G), then f is G-outer [35, p. 68]. One might conjecture that if f ∈ H 2 (G) is G-outer, then [ f ] = M(ρ ),

where ρ = f + / f − . However this is not the case.

Chapter 7. Cyclic invariant subspaces

82

Example 7.2.4. Consider the G-outer function f (z) = z. By (7.3.1) and Theorem 7.3.2 (see below) [z] = {g ∈ M(1) : g(0) = 0} which is a proper subset of M(1). Remark 7.2.5. One might wonder where the classical Hardy space H 2 (D) fits in with M(1). They do look very similar. It follows from (2.1.4) that H 2 (D) ⊂ M(1) with continuous embedding. However, this containment is proper. For example, the function 1 f (z) = √ 1−z is analytic across [0, 1] but does not belong to H 2 (D) since k f k2H 2 (D)

=

Z 2π 0

dθ 1 = ∞. |1 − eiθ | 2π

However, f ∈ N + (G) and by (6.2.5) f |∂ G ∈ L2 (∂ G, ω ∗ ). Thus, by Proposition 2.4.10, f ∈ H 2 (G) and hence, since f is analytic on D, f ∈ M(1).

Corollary 7.2.6. Suppose f = Θ f1 , where Θ is G-inner and f1 is G-outer such that f1 and 1/ f1 belong to H 2 (G). Then [ f ] = Θ · M(ρ ), where ρ = f1+ / f1− .

Proof. Use the fact that Θ is G-inner and so multiplication by Θ is an isometry to argue that [ f ] = Θ · [ f1 ]. Now use Theorem 7.2.2.

7.3 Polynomial approximation Our results have applications to polynomial approximation and analytic bounded point evaluations. Let ω be harmonic measure for ∂ G and P2 (ω ) be the closure of the analytic polynomials in L2 (ω ). Notice that [1] = P2 (ω ) ⊂ M(1). Since d ω |T ≍ |ψ ′ |dm, where ψ is the conformal map from G onto D, and since θ → is a log-integrable bounded function on [0, 2π ] (see (6.2.5)), there is a bounded D-outer function F on D such that |ψ ′ | = |F|2 almost everywhere on T. For an analytic polynomial p we can apply the Cauchy integral formula (see (2.4.11)) to see that for fixed a ∈ D, Z p(ζ )F(ζ ) p(a)F(a) = dm(ζ ). T 1 − ζa

ψ ′ (eiθ )

7.3. Polynomial approximation

83

The Cauchy-Schwarz inequality yields |p(a)F(a)| 6 Ca 6 Ca

Z

T

Z

T

2

2

|p| |F| dm |p|2 d ω

6 Ca kpkL2 (ω ) .

1/2

1/2

Divide through by F(a) (which is never zero since F is D-outer) to get |p(a)| 6 ca kpkL2 (ω )

(7.3.1)

for all polynomials p. In other words, D is the set of bounded point evaluationsfor P2 (ω ). This also means that the linear functional p 7→ p(a), initially defined on the analytic polynomials, continues to a bounded linear functional on P2 (ω ). Theorem 7.3.2. For f ∈ H 2 (G), the following are equivalent. 1. f ∈ P2 (ω );

2. f ∈ M(1); 3. f has an analytic continuation to D; 4. f ∈ closH 2 (G) H ∞ (D). Proof. The equality [1] = P2 (ω ) is clear. Theorem 7.2.2 gives us [1] = M(1) and so (1) ⇔ (2). From the proof of Proposition 6.1.1 we have [1] = closH 2 (G) H ∞ (D) b \ γ along with Corollary 4.2.20 and so (1) ⇔ (4). Now use the conformal map α : G → C to see that (3) ⇔ (4).

Chapter 8

The essential spectrum 8.1 Fredholm theory If B(H) is the algebra of bounded linear operators on a Hilbert space H and K is the ideal of compact operators on H, one forms the Calkin algebra B(H)/K and the natural map π : B(H) → B(H)/K. Recall that A ∈ B(H) is Fredholm if π (A) is invertible in B(H)/K. A well-known theorem [20, p. 356] says that A is Fredholm precisely when RngA is closed and both ker A and H/RngA are finite dimensional. An operator A is semi-Fredholm if π (A) is either right or left invertible in B(H)/K. Equivalently, A is semi-Fredholm if and only if Rng(A) is closed and either ker(A) or H/RngA is finite dimensional. We also use the notation

σ (A) := {λ ∈ C : λ I − A is not invertible} (spectrum of A), σe (A) := {λ ∈ C : λ I − A is not Fredholm} (essential spectrum of A). Note that σe (A) ⊂ σ (A). For a semi-Fredholm operator A let ind(A) := dim ker A − dim(H/RngA) be the index of A. When the set Z ∪ {±∞} is endowed with the discrete topology, the map A 7→ ind(A) (from the set of semi-Fredholm operators to Z ∪ {±∞}) is continuous [20, p. 361].

8.2 Essential spectrum We now compute the essential spectrum of T := S|M,

Chapter 8. The essential spectrum

86

where S is, as always, S f = z f on H 2 (G), and M is a non-zero invariant subspace for S. For λ ∈ G, it is easy to show that (S − λ I)H 2(G) = { f ∈ H 2 (G) : f (λ ) = 0} and thus is a (closed) non-trivial subspace of H 2 (G). Furthermore, ker(S − λ I) = {0}. From here it follows that σ (S) = D− and σe (S) ⊂ ∂ G. A result from [21, Thm. 4.3] proves the other inequality and so

σe (S) = ∂ G. For each λ ∈ G there is a cλ > 0 so that k(z − λ ) f k > cλ k f k

∀ f ∈ H 2 (G)

and so this same inequality holds for all f ∈ M. This inequality says that T − λ I has closed range. Clearly ker(T − λ I) = {0}. For λ ∈ G \ Z(M), we can use the nearly invariance of M (Corollary 6.1.3) to get that (T − λ I)M = { f ∈ M : f (λ ) = 0},

which is closed. Furthermore, M/(T − λ I)M is one-dimensional1. From here it follows that σ (T ) = D− . By our discussion above, T − λ I is Fredholm for all λ ∈ G \ Z(M) and ind(T − λ I) := dim ker(T − λ I) − dim(M/(T − λ I)M) = −1 ∀λ ∈ G \ Z(M). From here2, one can use the fact that Z(M) is a discrete set to show that (T − λ I) is Fredholm for all λ ∈ G and so σe (T ) ⊂ ∂ G. (8.2.1) In addition,

ind(T − λ I) = −1 ∀λ ∈ G.

(8.2.2)

σe (T ) = σl (T ),

(8.2.3)

Using standard Fredholm theory3 and the above identity on the index, we have

where σl (T ) is the left spectrum (also known as the approximate point spectrum) of T . It is a standard fact [20, p. 215] that ∂ σ (T ) ⊂ σl (T ) and so, since σ (T ) = D− , we get T ⊂ σe (T ). 1 See

also (see [60, Lemma 2.1]) are using the following general fact [22, p. 357]: Suppose A ∈ B(H) is Fredholm. Then there is an ε > 0 such that if Y ∈ B(H) with kY k (the operator norm of Y ) less than ε , then A + Y is also Fredholm and ind(A +Y ) = ind(A). 3 Combine Proposition 4.3, Proposition 4.4, and Proposition 6.10 from [22]. 2 We


87

Combine this with (8.2.1) and (8.2.3) to obtain T ⊂ σe (T ) = σl (T ) ⊂ ∂ G.

(8.2.4)

Thus to determine σe (T ), it remains to determine which points in [0, 1) belong to σe (T ). Theorem 8.2.5. Let M be a non-zero invariant subspace of H 2 (G) and let A(M) be the set of points x ∈ [0, 1) with the property that there exists an fx ∈ M such that f / fx extends to be analytic in a neighborhood of x whenever f ∈ M. Then with T := S|M we have

σe (T ) = ∂ G \ A(M). Proof. Let x ∈ A(M). For y ∈ G \ (Z(M) ∪ fx−1 ({0})) and close to x we know, since M is nearly invariant (Corollary 6.1.3), that f−

f f x (y) f x

z−y

∈ M ∀ f ∈ M.

Since x ∈ A(M) we can let y → x to obtain R f :=

f−

f f x (x) f x

z−x

∈M

∀ f ∈ M.

To show that R is continuous on M we will use the closed graph theorem. Indeed suppose ( fn )n>1 ⊂ M with fn → f and R fn → g in the norm of H 2 (G). Note that g ∈ M and fn → f , R fn → g pointwise in G. A little algebra shows that fn f g (x) → (z) − (z − x) (z) ∀z ∈ G \ fx−1 ({0}). fx fx fx Since f , g ∈ M, the function on the right is analytic near x and equal to ( f / fx )(x) when z = x. Thus fn f (x) → (x) fx fx which says that g = R f and so, by the closed graph theorem, R is continuous. A routine computation will show that R(T − xI) = I. Thus (T − xI) is left-invertible. But since T satisfies σl (T ) = σe (T ) (see (8.2.3)) we see that x 6∈ σe (T ). Conversely, suppose that x ∈ [0, 1) \ σe (T ). Then, from (8.2.3), x 6∈ σl (T ) and so T − xI has a left inverse R (which we will show in a moment is equal to the operator R from the previous paragraph). Since R(T − xI) = I we see that Rng(R) = M. Using the fact that x 6∈ σe (T ) and σe (T ) ⊂ ∂ G (see (8.2.1)), we know, from (8.2.2) and the continuity of the index [20, p. 361], that ind(T − xI) = lim ind(T − (x + iε )I) = −1. ε →0+


88 Furthermore [20, p. 363], 0 = = = =

ind(R(T − xI))

ind(R) + ind(T − xI) dim(ker R) − dim(RngR)⊥ − 1

dim(ker R) − 1 (since Rng(R) = M).

Thus ker R = C fx

(8.2.6)

for some fx ∈ M \ {0}. For y in some open neighborhood of x, consider the operator Ry := R(I − (y − x)R)−1. A computation using the identity ∞

Ry =

∑ (y − x)nRn+1

(8.2.7)

n=0

(which is valid for y in some small open neighborhood of x) shows that Ry (T − yI) = Ry ((T − xI) + (x − y)I) = I. Moreover, for y ∈ G \ (Z(M) ∪ fx−1 ({0})) and near x, the operator Qy f :=

f−

f f x (y) f x

z−y

(8.2.8)

is a bounded operator on M (again using the nearly invariance of M and the closed graph theorem). A computation will show this operator is also a left inverse for T − yI. Since y ∈ G \ (Z(M) ∪ fx−1({0})), we can use the nearly invariance of M, along with the facts (T − yI)M = { f ∈ M : f (y) = 0}; dim(M/(T − yI)M) = 1; fx (y) 6= 0, to see that M = (T − yI)M + C fx. Since Qy (T − yI) = Ry (T − yI) = I, then Qy = Ry on (T − yI)M. Furthermore, Qy fx = Ry fx = 0 (This follows from (8.2.8), (8.2.7), and (8.2.6)). Thus Qy = Ry on all of M and so f − ffx (y) fx Ry f = ∀ f ∈ M. (8.2.9) z−y


89

Finally, the identity in (8.2.7) shows that the function y 7→ Ry is an operator-valued analytic function for all y in some open neighborhood of x and so for fixed z0 ∈ G and f ∈ M y → Ry f (z0 ) is analytic in some open neighborhood of x. Choosing z0 such that fx (z0 ) 6= 0 and using the identity in (8.2.9), we see that f / fx is analytic in some open neighborhood of x. Thus x ∈ A(M) which completes the proof. Remark 8.2.10. Analytic continuation across boundary points seems to be a reoccurring theme when studying the essential spectra of multiplication (and Toeplitz) operators on certain Banach spaces of analytic functions [9, 11, 21]. Corollary 8.2.11. If f ∈ H 2 (G) \ {0} and T := S|[ f ], then the following hold. 1. The function h/ f extends to be analytic on D for every h ∈ M. 2. σe (T ) = T.

Proof. Notice how statement (2) follows immediately from statement (1) since (1) shows that A([ f ]) = [0, 1). To prove (1) fix an h ∈ [ f ]. By the definition of [ f ], there is a sequence of analytic polynomials (pn )n>1 such that pn f → h in the norm of H 2 (G). To show that h/ f has an analytic continuation across [0, 1), and thus complete the proof of (1), we will show that the sequence (pn )n>1 forms a normal family on D. Notice from Proposition 2.4.13 how (pn )n>1 forms a normal family on G. To this end, fix r ∈ (0, 1) and let Cr := {|z| = r}. We will assume that r is chosen so that f is non-zero on Cr and that both f + (r) and f − (r) exist and are non-zero. It follows that 0 < m 6 | f (z)| 6 M < ∞ ∀z ∈ Cr ∩ G. For a compact set A ⊂ rD we can apply the Cauchy integral formula (with an appropriate branch cut for the square root) to get (z − r)1/2 pn (z) =

1 2π i

I

Cr

(η − r)1/2 pn (η ) dη η −z

∀z ∈ A.

Now apply the following three inequalities |pn (η )| 6

1 |pn (η ) f (η )|, m

|pn (η ) f (η )| 6 K 4

η ∈ Cr ∩ G;

kpn f k , dist(η , ∂ G)1/2

This inequality follows from Proposition 2.4.13 and (6.2.5).

η ∈ Cr ∩ G4 ;


90 |η − r|1/2 6 K, dist(η , ∂ G)1/2 to the above integral identity to show that

η ∈ Cr ∩ G

|z − r|1/2 |pn (z)| 6 Kkpn f k

∀z ∈ A.

But since kpn f k is uniformly bounded in n and since z ∈ A and A is a compact subset of rD, we see that |pn (z)| 6 K ∀z ∈ A, ∀n > 1. It follows that the sequence (pn )n>1 forms a normal family on D.

2

Corollary 8.2.12. Let M be an invariant subspace of H (G) and T = S|M. If x ∈ [0, 1) is a cluster point for the zeros or poles of f /g for some f , g ∈ M \ {0}, then x ∈ σe (T ).

Proof. Assume to the contrary that x 6∈ σe (T ). Then, by Theorem 8.2.5, there is a function fx ∈ M such that h/ fx extends to be analytic in a neighborhood of x for every h ∈ M. In particular, f / fx and g/ fx extend to be analytic near x. Depending on the orders of the zeros of these two functions at x, either f /g or g/ f extend to be analytic near x. This contradicts the fact that x is an accumulation point for the zeros (or poles) of f /g. From Theorem 7.2.2 we know, for certain f , that [ f ] = M(ρ , [0, 1]), where ρ = f + / f − . From here, one might be tempted to conclude that spaces of the form M(ρ , [0, 1]), for some measurable ρ : [0, 1] → C, are always cyclic. The following example shows that this is not always the case. Example 8.2.13. There are two G-inner functions f , g and a measurable function ρ : [0, 1] → C such that if M := [ f , g], the invariant subspace generated by f and g, then the following hold: 1. M ⊆ M(ρ , [0, 1]).

2. σe (S|M) = σe (S|M(ρ , [0, 1]) = ∂ G. 3. S|M and S|M(ρ , [0, 1]) are not cyclic. Proof. Choose a sequence (an )n>1 ⊆ D+ such that (an )n>1 clusters precisely on all of [0, 1] and (an )n>1 is an H 2 (G) zero set. Let f be a G-Blaschke product whose zero set is precisely (an )n>1 . Since (an )n>1 is also an H 2 (D+ ) zero set, then there is also a D+ Blaschke product b whose zeros are precisely (an )n>1 . Notice that b extends to be analytic across (−1, 0) since the zeros of b do not accumulate (−1, 0). Furthermore, since |b(x)|2 = 1 almost everywhere on [−1, 1], the analytic continuation of b across (−1, 0) is the function 1/b∗ where b∗ is the D− -inner function b∗ (z) = b(z). Define a function g as follows: f (z)/b(z), if z ∈ D+ ; g(z) = f (z)b∗ (z), if z ∈ D− .

Notice that g ∈ H ∞ (G) and that g is a G-inner function. Since b is D+ -inner we also have g+ (x) f + (x)/b(x) f + (x)b∗ (x) f + (x) = − = − = − − ∗ ∗ g (x) f (x)b (x) f (x)b (x) f (x)


91

for almost every x ∈ (0, 1). Thus if we set

ρ = g+ /g− = f + / f − , then ρ : [0, 1] → C is a measurable function and f , g ∈ M(ρ , [0, 1]), so M := [ f , g] ⊆ M(ρ , [0, 1]). Since f /g has infinitely many zeros and poles clustering on [0, 1], we have the containment [0, 1] ⊆ σe (S|M) (Corollary 8.2.12). Similarly, since f , g ∈ M(ρ , [0, 1]), we see that [0, 1] ⊆ σe (S|M(ρ , [0, 1]). Thus, from (8.2.4), we have σe (S|M) = ∂ G and

σe (S|M(ρ , [0, 1]) = ∂ G. Hence, by Corollary 8.2.11, neither S|M nor S|M(ρ , [0, 1]) is cyclic.

Chapter 9

Other applications 9.1 Compressions We now examine the compression of S to certain co-invariant subspaces. Throughout this section ω will denote harmonic measure for ∂ G at some point in G. Note from (6.2.5) that d ω ≍ |ξ |−1/2|ξ − 1|ds. We begin with the following. Proposition 9.1.1. The map R : H 2 (G) → L2 ([0, 1], ω ) defined by R f = f + − f − is a continuous onto linear operator.

Proof. The obvious estimates will show that R is continuous. Let φ = φG : D → G from the appendix and ψ = φ −1 . To show R is onto, let g ∈ L2 ([0, 1], ω ) and note that g(x) = k ◦ ψ (x) for some k ∈ L2 (J, d θ ), where J = {eiθ : 0 6 θ 6 π /2} and ψ + ([0, 1]) = J. In other words, we are thinking of g as living on the ”top part” of the slit [0, 1]. The function k(eiθ ), extended to be zero for θ ∈ [π /2, π ], has a Fourier sine series ∞

k∼ where an =

2 π

∑ an sin(nθ ),

n=1

Z π /2

k(eit ) sin(nt)dt.

0

Define h(z) :=

1 ∞ ∑ anzn 2i n=1

and notice that h ∈ H 2 (D) (since the an ’s are square summable - see (2.1.8)) and h(eiθ ) − h(e−iθ ) = k(eiθ ),

a.e. θ ∈ [0, π /2].

Chapter 9. Other applications

94

Now let f := h ◦ ψ and see that f ∈ H 2 (G) (since f ◦ φ = h ∈ H 2 (D)) and for almost every x ∈ [0, 1], f + (x) − f − (x) = h(eiθ ) − h(e−iθ ) = k(eiθ ) = (k ◦ ψ )(x) = g(x).

Thus R is onto. Notice that, ker(R) = M(1). This allows us to define the quotient operator Re : H 2 (G)/M(1) → L2 ([0, 1], ω ),

Re fe := R f = f + − f − ,

where fe is the coset in H 2 (G)/M(1) represented by f . With S defined as multiplication by z on H 2 (G), one forms the bounded operator Se : H 2 (G)/M(1) → H 2 (G)/M(1),

An easy calculation shows that

e ReSe = Mx R,

Sefe := zef . (9.1.2)

where Mx g = xg is multiplication by x on L2 ([0, 1], ω ). Putting the above discussion in more appropriate Hilbert space language yields the following. Proposition 9.1.3. Suppose N is the co-invariant subspace N = H 2 (G) ⊖ M(1), PN is the orthogonal projection of H 2 (G) onto N, and C = PN S|N is the compression of S to N. Then C is similar to Mx on L2 ([0, 1], ω ). Corollary 9.1.4. If h, 1/h ∈ H ∞(G) and ρ = h+ /h−, then the compression of S to H 2 (G)⊖ M(ρ ) is similar to Mx on L2 ([0, 1], ω ) Proof. Let A = RM1/h where R is the map from Proposition 9.1.1 and the operator M1/h : H 2 (G) → H 2 (G) is defined by M1/h f = f /h. One easily sees that A : H 2 (G) → L2 ([0, 1], ω ) is onto, intertwines S with Mx , and the kernel of A is precisely M(ρ ). The result follows by passing to quotients as above. Using Wiener’s theorem [44, p. 7] we know that every Mx -invariant subspace of L2 ([0, 1], ω ) is of the form χE c L2 ([0, 1], ω ) for some measurable set E ⊂ [0, 1]. Bringing in Theorem 7.2.2 and (9.1.2), we can prove the following. Theorem 9.1.5. Suppose M is an invariant subspace of H 2 (G) and there exists an f ∈ M such that f and 1/ f belong to H ∞ (G). Then there is a measurable set E ⊂ [0, 1] such that M = M(ρ , E), where ρ = f + / f − . Remark 9.1.6. Compare this to Theorem 7.2.2.

9.2. The parameters

95

9.2 The parameters Let us focus some more on the invariant subspaces M(ρ ) := f ∈ H 2 (G) : f + = ρ f − a.e. on [0, 1] ,

where ρ is a complex-valued measurable function on [0, 1]. We ask the question: For what measurable ρ is M(ρ ) 6= {0}? Let us first discuss a necessary condition. We see from (2.4.3) that for F ∈ H 2 (G) \ {0}, Z Z | log |F + || + | log|F − || d ω + | log |F||d ω < ∞. [0,1]

T

In particular, if ρ : [0, 1] → C is measurable and F ∈ M(ρ ) \ {0}, we can combine the above observation with the identity ρ = F + /F − almost everywhere to see the following. Proposition 9.2.1. Suppose ρ : [0, 1] → C is measurable and M(ρ ) 6= {0}, then log |ρ | ∈ L1 ([0, 1], ω ). Is the necessary condition in the above proposition sufficient? Let us work through a few examples.

Proposition 9.2.2. If ρ is a complex-valued step function which is never zero on [0, 1], then M(ρ ) 6= {0}. Proof. Let p0 , p1 , · · · , pn be non-zero complex numbers such that ℜp j > 0 for all j and 0 6 a1 < a2 < · · · < an < 1. Let f (z) := z p0 (z − a1 ) p1 · · · (z − an ) pn ,

where we take the branch of the complex logarithm to be analytic on C \ [0, ∞). Notice, since ℜp j > 0, that f ∈ H ∞ (G) \ {0}. Then with

ρ (x) := e2π ip0 ,

0 < x < a1 ,

ρ (x) := e2π ip0 e2π ip1 , and so on, we see that

a0 < x < a1 ,

f+ = ρ f−

and so M(ρ ) 6= {0}. By choosing appropriate a j and p j one can create any non-vanishing step function ρ . Proposition 9.2.3. Suppose ρ is a real-valued measurable function on [0, 1] such that a 6 ρ (x) 6 b,

x ∈ [0, 1]

for some constants 0 < a < b < ∞. Then M(ρ ) 6= {0}.

(9.2.4)

Chapter 9. Other applications

96 Proof. Let h(z) :=

1 2π i

Z 1 log ρ (t)

t −z

0

dt

be the Borel transform of the function χ[0,1] log ρ and notice that h is analytic on C \ [0, 1]. It is well-known that the limits h+ (x) and h− (x) exist for almost every x ∈ R (see [71, Theorem 1.4] for details). A computation shows that h(x + iy) − h(x − iy) = where Px+iy (t) =

Z 1 0

Px+iy (t) log ρ (t)dt,

1 y π (t − x)2 + y2

is the usual Poisson kernel for the upper half plane. We will allow negative values of y in the above formula. Using standard facts about Poisson integrals [35, p. 29], we get h+ − h− = χ[0,1] log ρ almost everywhere. The identities Px+iy (t) = −Px−iy (t) and

1 ℜh(x + iy) = 2

Z 1 0

Px+iy (t) log ρ (t)dt

along with (9.2.4) will show that ℜh is a bounded function on G. Thus f := eh belongs to H ∞ (G) \ {0} and f + = ρ f − and so f ∈ M(ρ ) \ {0}.

Chapter 10

Domains with several slits 10.1 Statement of the result We now extend our main theorem (Theorem 6.2.1) to domains with several slits. More precisely, we consider domains of the form G = D\

N [

γ j,

(10.1.1)

j=1

where γ j are analytic arcs satisfying certain technical conditions. They are the following (see Figure 10.1 for an example): 1. Each γ j is a closed connected subset of a simple analytic open arc γbj which meets T at a positive angle. The arc γbj will be called an analytic continuation of γ j ;

2. γ1 , · · · , γN are pairwise disjoint;

3. Each γ j has one end point λ j ∈ T and γ j \ {λ j } ⊂ D. Our extension of Theorem 6.2.1 is the following.

Theorem 10.1.2. Let G be a domain as in (10.1.1) and let γbj , 1 6 j 6 N, be analytic continuations of the arcs γ j , 1 6 j 6 N. If M is a non-trivial invariant subspace of H 2 (G) with greatest common G-inner divisor ΘM , then there is a measurable set E⊂

N [

γj,

j=1

a measurable function ρ : E → C, and an analytic function F on an open set V ⊂ G, with N [

j=1

γbj ∩ D ⊂ V,

Chapter 10. Domains with several slits

98

λj

γbj

γj

0

γk λk γbk

Figure 10.1: A domain as in (10.1.1) with several slits. Notice how each slit γ j is part of a larger analytic slit γbj and how the slits meet the circle at a positive angle. such that

M = ΘM ·

f ∈ H 2 (G) :

f |V ∈ H 2 (V ), f + = ρ f − a.e. on E . F

Remark 10.1.3. 1. The functions f + and f − are the non-tangential limits of f from opposite sides (once an orientation is fixed) of the arc γ j . 2. The open set V will turn out to be V=

N [

(V j ∩ G),

j=1

where V j are certain disjoint domains obtained from Lemma 10.2.1 (below). See also Figure 10.2. 3. Since V is a disjoint union of simply connected domains, we should be clear what we mean by H 2 (V ). We follow [21, p. 664]. An analytic function f on V belongs to H 2 (V ) if | f |2 has a harmonic majorant on V . We let U f denote the least harmonic majorant and for a j ∈ V j ∩ G, 1 6 j 6 N, define the norm on H 2 (V ) to be k f k2H 2 (V ) :=

N

∑ U f (a j ).

j=1

As to be expected, different choices of a j ’s yield equivalent norms.

10.2. Some technical lemmas

99

b 4. It should be the case, as in Theorem 6.2.1, that there is a G-outer function F, where b = D\ G

such that M = ΘM ·

f ∈ H 2 (G) :

N [

(γbj )− ,

j=1

f b f + = ρ f − a.e. on E . ∈ H 2 (G), F

However, at this point, we do not see how to produce this more global outer function from the local function we have in Theorem 10.1.2.

10.2 Some technical lemmas We will use Theorem 6.2.1 to prove Theorem 10.1.2. In order to do this, we need to take care of a few technicalities. Lemma 10.2.1. There are open subsets V1 , · · · ,VN of D such that 1. V1 , · · · ,VN are pairwise disjoint.

2. For each j, V j contains γ j \ {λ j }.

3. For each j, ∂ V j \ {λ j } is a C2 arc and ∂ V j is a piecewise C2 arc which intersects T only at λ j and, at this point, makes a positive angle with both γ j and T.

4. For each j, there is a conformal map β j from V j onto D such that

β j (γ j \ {λ j }) = [0, 1).

1 0 11 00 00 11

λj

γj

1 0 0 1 0

Vj

11 00 00 1 11

βj

Figure 10.2: The domain V j (shaded) and the conformal map β j : V j → D which satisfies β j (γ j \ {λ j }) = [0, 1).


100

Proof of Lemma 10.2.1. Fix j = 1, · · · , N and let γ = γ j , γb = γbj , and λ = λ j . By the definition of analytic arc, there is an open set U of γb and a conformal map α which maps U onto a region R which is symmetric about an open interval (a, b) which contains [0, 1]. We can assume that α (γ ) = [0, 1] and α (λ ) = {1}. By shrinking U, we can also assume that ∂ R is analytic (see Figure 10.3).

α

λ U

γb

11 00 00 11

γ

R a

b 0

1

Figure 10.3: The region U (shaded) and R = α (U). Note that α (γ ) = [0, 1] with α (λ ) = {1}. The curve α (U ∩ T) will be an analytic arc in R that passes through the point 1. Pick two points z1 , z2 ∈ ∂ R which are symmetric about [a, b] and such that the line segments ℓ1 , ℓ2 (which connect z1 , respectively z2 , to 1) only intersect Y := α (U ∩ T) at 1 and form a positive acute angle to both [a, b] and Y at 1. Now form the region R1 bounded by the line segment [a, 1], the line ℓ1 , and the part of ∂ R subtended by z1 and a (see Figure 10.4). z1 1 0 0 1 R1

Y

ℓ1

a 0

ℓ2

1

b

00 11 z2

Figure 10.4: The region R1 (top). By altering the definition of R1 slightly (replacing part of the line segments ℓ1 and ℓ2 with appropriate C2 curves near the points z1 and z2 ) we can assume that ∂ R1 \ {1} is C2 . One can find a conformal map α1 from R1 onto D+ such that

α1 (a) = −1,

α1 (0) = 0,

α1 (1) = 1

10.3. A localization of Yakubovich

101

(see [56, p. 319]). By the symmetry principle, α1 (really the analytic continuation of α1 ) will map R1 ∪ (a, 1) ∪ {z : z ∈ R1 } onto D with α1 ([0, 1)) = [0, 1). Therefore the map α −1 ◦ α1−1 will map D onto a region V = V j with the desired properties (2) and (3). Finally, let β = β j be the inverse of α −1 ◦ α1−1 .

10.3 A localization of Yakubovich The proof of Theorem 10.1.2 depends on this following localization result discovered by Yakubovich [79, Lemma 4] in the case of an annular domain. Proposition 10.3.1. Suppose that G is a multiple slit domain as in (10.1.1) and let V :=

N [

(V j ∩ G).

j=1

If M is a non-trivial invariant subspace of H 2 (G) with greatest common G-inner divisor equal to one, then f ∈ H 2 (G) belongs to M if and only if f |V belongs to the closure of M|V in H 2 (V ). To make writing integrals more manageable, we will let ω be harmonic measure for G at φ (0), where φ : D → G and ψ = φ −1 . From our discussion of harmonic measure from Chapter 2, note that

ω

N

=

∑ (ω0,D ◦ ψ +j + ω0,D ◦ ψ −j ) + ω0,D ◦ ψ |T

j=1

|(ψ ′ )+j |

N

=

∑

2π

j=1

ds +

|(ψ ′ )−j | 2π

!

ds + |ψ ′ |

dθ . 2π

In the above, (ψ ′ )+j and (ψ ′ )−j denote the upper and lower boundary functions for ψ ′ on

γ j . For h ∈ H 2 (G) · H 2 (G), we will use the notation Z

∂G

hd ω := ∗

N

∑

j=1

Z

γj

h+j

|(ψ ′ )+j | 2π

ds +

Z

γj

h−j

|(ψ ′ )−j | 2π

!

ds +

Z

T

h|ψ ′ |

dθ . 2π

(10.3.2)

We invite the reader to verify the following Cauchy-Schwarz type inequality Z

∂G

| f g|d ω 6 (2N + 1) ∗

Z

∂G

| f | dω 2

∗

1/2 Z

∂G

|g| d ω 2

∗

1/2

.

(10.3.3)

Needed in the proof of Proposition 10.3.1 will be the following discussion of the growth (and decay) of inner and outer functions. If S is a singular D-inner function with an atom at ζ0 ∈ T, then c |S(rζ0 )| = O(e− 1−r ), r → 1−


102 for some c > 0. For g ∈ L1 (m) we have Z

T

Prζ0 (ξ )|g(ξ )|dm(ξ ) = o(

1 ), 1−r

r → 1− .

(10.3.4)

Certainly (10.3.4) is true when g is continuous. To see the general case, approximate g with continuous functions in the L1 (m) norm and use the identity sup Prζ0 (ξ ) =

ξ ∈T

1+r . 1−r

This means that if H is a D-outer function, then Z ξ + rζ0 H(rζ0 ) = exp log |H(ξ )|dm(ξ ) T ξ − rζ0 and so, since ℜ

ξ + rζ0 = Prζ0 (ξ ) ξ − rζ0

and log |H| ∈ L1 (m), we have, via (10.3.4),

cr 1 6 e 1−r , |H(rζ0 )|

where cr → 0 as r → 1− . From here we see that S(rζ0 ) = 0. lim r→1− H(rζ0 )

(10.3.5)

See [41] for a related result. We are now ready for the proof of Proposition 10.3.1. Proof of Proposition 10.3.1. Recall that the regions V j are from Lemma 10.2.1 and the open set V is N [

(V j ∩ G).

j=1

It is obvious that if f ∈ M, then f |V ∈ M|V . So suppose that g ∈ H 2 (G) and g|V = lim fn |V, n→∞

(10.3.6)

where fn ∈ M and the limit in (10.3.6) is in the norm of H 2 (V ). Our goal is to show that g ∈ M. Recall that λ j is the endpoint of γ j which lies on T. Define a polynomial Q by N

Q(z) = ∏ (z − λ j )b , j=1


103

where b is a positive integer large enough so that |Q|d ω ∗ 6 cd ωV∗ and |Q|ds 6 cd ωV∗

S

on

on

S

j γ j;

(10.3.7)

j ∂Vj.

(10.3.8)

Here is where we use the hypothesis that ∂ V j \ {λ j } is C2 and ∂ V j is a piecewise C2 arc which meets T and γ j at positive angles. See (2.3.14) for an explanation of this. The rather lengthy argument below will show, for each h ∈ M⊥ , that Qg , h = 0, |λ | > 1. (10.3.9) z−λ Assuming this is true, let us see how prove that g ∈ M. Using (10.3.9) and Runge’s theorem [19, p. 198] we see that Qg ⊥ h for all h ∈ M⊥ . Hence Qg ∈ M. We now need to argue that g ∈ M. Since the zeros of the polynomial Q lie on T, Q is D-outer and so [35, p. 85] we can choose (Qn )n>1 ⊂ H ∞ (D) such that Qn Q → 1 weak-∗ in H ∞ (D). Since M is H ∞ (D)-invariant (Proposition 6.1.1), we see that Qn Qg ∈ M. Furthermore, Qn Qg → g weakly in H 2 (G) and so g ∈ M. Before getting to the heart of the proof of (10.3.9), we first need to derive a few integral formulas. For f ∈ M \ {0} and h ∈ M⊥ , note, since M is invariant, that f , h = 0, |λ | > 1. (10.3.10) z−λ If w(ζ ) := |ψ ′ (ζ )|,

ζ ∈ T,

an application of Fatou’s jump theorem (Theorem 3.3.3) implies lim

Z

r→1− ∂ G

fh d ω ∗ = ζ f (ζ )h(ζ )w(ζ ), z − rζ

a.e. ζ ∈ T.

(10.3.11)

Recall our notational understanding discussed just before this proof (see (10.3.2)) and notice in (10.3.11) how the integrals over the slits cancel out in the limit. Define h1 (λ ) :=

1 f (λ )

Z

∂G

fh dω ∗, z−λ

λ ∈ G \ f −1 ({0}).

(10.3.12)

From (10.3.11) we get lim h1 (rζ ) = ζ h(ζ )w(ζ ),

r→1−

a.e. ζ ∈ T.

(10.3.13)

This shows, via uniqueness of radial limits of quotients of H p (G) functions, that h1 is independent of f ∈ M \ {0}. Thus, since the greatest common G-inner divisor of M is


104

one, we conclude that h1 , initially defined on G \ f −1 ({0}), has an analytic continuation to G and furthermore, h1 ∈ N + (G). Note that Q f ∈ M and so from (10.3.12) we have Q(λ ) f (λ )h1 (λ ) =

Z

∂G

Qfh d ω ∗. z−λ

(10.3.14)

Since the curves ∂ V j , γ j , T all meet at λ j at positive angles to each other, we see that for fixed λ ∈ ∂ V j \ {λ j } 1 1 ≍ , sup |λ − λ j | ζ ∈T |ζ − λ | sup z∈γ j

1 1 ≍ . |z − λ | |λ − λ j |

From these estimates, (10.3.14), and the Cauchy-Schwarz type inequality from (10.3.3) it follows that Q f h1 is a bounded function on each ∂ V j and ( ) sup |Q(λ ) f (λ )h1 (λ )| : λ ∈

N [

∂Vj

j=1

6 ck f kH 2 (G) ,

f ∈ M,

(10.3.15)

where c > 0 is independent of f . For general g ∈ H 2 (G) (not necessarily in M), define ℓ(λ , g) := Z

Z

Qgh d ω ∗, z ∂G − λ

λ ∈ De ,

Qgh d ω ∗ − Q(λ )g(λ )h1(λ ), λ ∈ G. ∂G z − λ An argument using Fatou’s jump theorem (Theorem 3.3.3) and (10.3.13) will show that ℓ(λ , g) :=

lim ℓ(rζ , g) = lim ℓ(sζ , g),

r→1−

s→1+

a.e. ζ ∈ T

(10.3.16)

and so the functions ℓ(·, g)|G and ℓ(·, g)|De are ‘pseudocontinuations’ of each other across T (see [64] for more on pseudocontinuations). Privalov’s uniqueness theorem says that pseudocontinuations are unique in that if L is another analytic function on G whose non-tangential boundary values are equal almost everywhere to those of ℓ(·, g)|De , then L = ℓ(·, g)|G. If g ∈ M, then by (10.3.10) ℓ(·, g) ≡ 0 on De and so, by uniqueness of pseudocontinuations, ℓ(·, g) ≡ 0 on G ∪ De for all g ∈ M. (10.3.17) For a compactly supported measure µ in C, the Cauchy transform (C µ )(λ ) :=

Z

1 d µ (z) z−λ


105

b \ supt(µ ) and is analytic on C |(C µ )(λ )| 6

1 kµ k, dist(λ , supt(µ ))

b \ supt(µ ), λ ∈C

(10.3.18)

∀g ∈ H 2 (G), λ ∈ K.

(10.3.19)

where kµ k is the total variation norm of µ . For a compact set K ⊂ G, we can use the estimate in (10.3.18), the standard continuity of point evaluations for H 2 (G) (Proposition 2.4.13), and (10.3.3) to obtain a constant CK > 0, depending only on K, with |ℓ(λ , g)| 6 CK kgkH 2 (G)

We will now derive, for certain g ∈ H 2 (G), a useful integral formula for ℓ(·, g). Define PV := g ∈ H 2 (G) : Qgh1 |(∪ j ∂ V j ) ∈ L1 (∪ j ∂ V j , ds)

and notice from (10.3.15) that

M ⊂ PV . Define For λ ∈ V and g ∈ PV , the function

(10.3.20)

Ω := G \ V − .

Hλ (z) :=

Q(z)g(z)h1 (z) z−λ

belongs to N + (Ω). Furthermore, by (10.3.13), Hλ (ζ ) =

Q(ζ )g(ζ )ζ h(ζ )w(ζ ) a.e. ζ ∈ T. ζ −λ

Observing that g|T, h|T ∈ L2 (wdm) we can use the Cauchy-Schwarz inequality to see that Hλ |T ∈ L1 (m). Using this observation along with our assumption that g ∈ PV we get that Hλ |∂ Ω ∈ L1 (∂ Ω, ds). Now apply Proposition 2.4.10 to see that Hλ ∈ E 1 (Ω). From Proposition 2.4.12, the familiar Cauchy’s theorem is now valid and so I

T

Hλ (z)dz −

I

∪ j ∂Vj

Hλ (z)dz =

I

∂Ω

Hλ (z)dz = 0,

(10.3.21)

where the orientation of the path integrals obeys the usual left-hand rule. Thus, for λ ∈ V and g ∈ PV , we have

= =

Z

Qgh d ω ∗ − Q(λ )g(λ )h1(λ ) z ∂G − λ I Z 1 Qgh1 Qgh dζ + d ω ∗ − Q(λ )g(λ )h1(λ ) (d ζ = iζ |d ζ | and (10.3.13)) 2π i T ζ − λ z ∪jγj − λ

ℓ(λ , g) =

1 2π i

I

∪ j∂Vj

Qgh1 dz + z−λ

Z

∪jγj

Qgh d ω ∗ − Q(λ )g(λ )h1(λ ) (by (10.3.21)) z−λ


106

For λ ∈ De and g ∈ PV , a similar computation with Cauchy’s theorem yields ℓ(λ , g) =

1 2π i

I

Qgh1 dz + z−λ

∪ j∂Vj

Z

∪ jγ j

Qgh dω ∗. z−λ

(10.3.22)

For λ ∈ G \ V − = Ω we have Z

Qgh dω ∗ T ζ −λ I 1 Qgh1 = d ζ (d ζ = iζ |d ζ | and (10.3.13)) 2π i T ζ − λ I I Qgh1 1 Qgh1 1 = dz + dz 2π i ∂ Ω z − λ 2π i ∪ j ∂ V j z − λ I 1 Qgh1 = Q(λ )g(λ )h1 (λ ) + dz (Cauchy’s formula - Proposition 2.4.12) 2π i ∪ j ∂ V j z − λ which says, for λ ∈ G \ V − , that 1 ℓ(λ , g) = 2π i

I

Qgh1 dz + z−λ

∪ j∂Vj

Z

∪ jγ j

Qgh dω ∗. z−λ

(10.3.23)

Combining (10.3.22) and (10.3.23) we see that ℓ|De and ℓ|(G \ V − ) are analytic continuations of each other. In summary, we have, for g ∈ PV , that ℓ(λ , g) (originally defined on b \ V − ) ∪V which satisfies the formulas De ∪ G) extends to be an analytic function on (C ℓ(λ , g) =

1 2π i

I

∪ j ∂Vj

Qgh1 dz + z−λ

Z

∪jγj

Qgh d ω ∗, z−λ

while ℓ(λ , g) =

1 2π i

I

∪ j∂Vj

Qgh1 dz + z−λ

Z

∪ jγ j

b \ V −, λ ∈C

Qgh d ω ∗ − Q(λ )g(λ )h1 (λ ), z−λ

(10.3.24)

λ ∈ V. (10.3.25)

Suppose Γ is a circle contained in G which intersects ∪ j ∂ V j in a finite set and such that the angle at each point of intersection with ∪ j ∂ V j is different from zero or π . We can apply our Cauchy transform estimate from (10.3.18), along with (10.3.3) and Proposition 2.4.13, to (10.3.24) and (10.3.25) to get Z dist(λ , Γ ∩ ∂ V )|ℓ(λ , g)| 6 AΓ kgkH 2 (G) + |Qgh1 |ds , λ ∈ Γ, g ∈ PV , ∪ j ∂Vj

(10.3.26) where AΓ > 0 depends only on Γ. Let σ be a G-inner function with atomic singularities at {λ1 , · · · , λN } (note that {λ j } = γ j ∩ T). From the fact that ∂ V j meets T at a positive angle we see, for every t > 0, that σ |∂ V j decreases to zero faster than any G-outer function on ∂ V j (see (10.3.5)). Thus,


107

for all t > 0, σ t h1 is a bounded function on ∂ V j for each j. This means that for any g ∈ H 2 (G) and t > 0 we have Z

∪ j∂Vj

Z

|Qgσ t h1 |ds 6

ct

6

ct

6 6

ct kgkH 2 (V ) ct kgkH 2 (G)

Hence

σ t g ∈ PV

∪ j ∂Vj

Z

∪ j ∂Vj

|Qg|ds |g|d ωV∗

(by (10.3.8))

(by (2.1.4)).

∀g ∈ H 2 (G),t > 0.

(10.3.27)

For each j = 1, · · · , N, let ∆ j be an open disk with ⊂ G and such that ∂ ∆ j ∩ ∂ V j is a finite set and the angle between ∂ ∆ j and ∂ V j is different from zero or π (see Figure 10.5). ∆−j

∆j

γj

∂Vj

Figure 10.5: The disks ∆ j . The facts that σ 1/n → 1 almost everywhere on ∂ G and |σ 1/n − 1| 6 2, along with the dominated convergence theorem and (10.3.19), will show that ℓ(λ , g) = lim ℓ(λ , σ 1/n g) uniformly on ∪ j ∂ ∆ j . n→∞

(10.3.28)

We pause to comment that (10.3.27) says that the functions σ 1/n f ( f ∈ M) and σ 1/n g (g ∈ H 2 (G)) can be applied to the integral formulas in (10.3.24) and (10.3.25) as well as the estimate in (10.3.26).


108

After this long preamble and getting things set up, we are now ready to complete the proof by verifying the identity in (10.3.9). If fn ∈ M approximate g as in (10.3.6), we can use the facts that |σ 1/n | 6 1, σ 1/n h1 is bounded on ∪ j ∂ V j , (10.3.8), and (10.3.3) to assume (by choosing an appropriate subsequence of the fn ’s) that Z lim kσ 1/n (g − fn)kH 2 (V ) + |Q(g − fn )σ 1/n h1 |ds = 0. ∪ j ∂Vj

n→∞

Apply this last limit identity to (10.3.26) to get lim dist(λ , ∂ ∆ j ∩ ∂ V j ) ℓ(λ , σ 1/n g) − ℓ(λ , σ 1/n fn ) = 0 n→∞

(10.3.29)

uniformly on ∂ ∆ j for all j. Our next tool will be a certain sequence of approximating polynomials (qn )n>1 . For each n ∈ N, choose open sub-arcs γ nj of γ j with one end point at λ j and such that lim ω ∗ ∪ j γ nj = 0. (10.3.30) n→∞

(see Figure 10.6)

γ nj

γj ∂Vj

Figure 10.6: The sub-arc γ nj and the set ∪ j (∂ V j ∪ γ j ) \ ∆ j . Let sn be a continuous function on [ j

(∂ V j ∪ γ j ) \ ∆ j

satisfying the three conditions |sn | 6 1,

sn = σ 1/n on

[ j

(∂ V j \ ∆ j ),

sn = 1 on

[ j

(γ j \ γ nj ).


109

Since the domain of definition of sn has connected complement, we can apply Lavrientiev’s theorem [23, p. 232] to produce an analytic polynomial qn satisfying |sn − qn| 6

1 . n(1 + k fnkH 2 (G) )

From here we get Z

6

Z

∪ j ∂ V j \∆ j ∪ j ∂ V j \∆ j

6 0+ 6

c . n

|(σ 1/n − qn)Q fn h1 |ds |(σ 1/n − sn )Q fn h1 |ds +

Z

∪ j ∂ V j \∆ j

|(sn − qn )Q fn h1 |ds

1 ck fn kH 2 (G) (by (10.3.15)) n(1 + k fnkH 2 (G) )

Moreover, Z

6

Z

∪ j γ j \γ nj ∪ j γ j \γ nj

|(σ 1/n − qn)Q fn h|d ω ∗ |σ 1/n − 1||Q fn h|d ω ∗ +

Z

∪ j γ j \γ nj

By (10.3.3) the first summand is bounded above by !1/2 Z Z 1/n 2 2 ∗ |σ − 1| |h| d ω c ∪ j γ j \γ nj

6

c

Z

∪ j γ j \γ nj

|σ

1/n

− 1| |h| d ω 2

2

∗

!1/2

|sn − qn||Q fn h|d ω ∗ .

|Q fn | d ω 2

∪ j γ j \γ nj

k fn kH 2 (V )

∗

!1/2

(by (10.3.7))

which converges to zero as n → ∞ since |σ 1/n − 1| 6 2, σ 1/n → 1 almost everywhere as n → ∞, and fn |V → g|V in H 2 (V ) norm. Again by (10.3.3) the second summand is bounded above by 1 ck fn kH 2 (G) n(1 + k fnkH 2 (G) ) which clearly converges to zero as n → ∞. Use the facts that ( ) sup sup |qn (λ )| : λ ∈ n

[ j

(∂ V j ∪ γ j ) \ ∆ j

< ∞ and

sup k fn kH 2 (V ) < ∞, n

along with (10.3.3) and (10.3.7) to get Z

∪ j γ nj

|σ 1/n − qn||Q fn h|d ω ∗ 6 c

Z

∪ j γ nj

|h|2 d ω ∗


110

which, by elementary measure theory and (10.3.30), goes to zero as n → ∞. Thus we have produced a sequence of analytic polynomials (qn )n>1 such that Z Z 1/n 1/n ∗ lim |(σ − qn)Q fn h|d ω = 0. (10.3.31) |(σ − qn)Q fn h1 |ds + n→∞

∪ j ∂ V j \∆ j

∪ jγ j

Combine (10.3.31) with (10.3.18) to get Z 1 I (σ 1/n − qn)Q fn h1 (σ 1/n − qn)Q fn h ∗ dz + dω dist(λ , ∂ V j ∩ ∂ ∆ j ) 2π i ∪ j ∂ V j \∆ j z−λ z−λ ∪jγj (10.3.32) goes to zero as n → ∞ uniformly on ∂ ∆ j for all j. Since M is an invariant subspace, qn fn ∈ M and so by (10.3.17) ℓ(·, qn fn ) ≡ 0 which means that ℓ(·, σ 1/n fn ) = ℓ(·, (σ 1/n − qn ) fn ). For λ ∈ G \ V − we can use (10.3.24) (applied to g := (σ 1/n − qn) fn which belongs to PV by (10.3.27) and (10.3.20)) to get ℓ(λ , (σ 1/n − qn ) fn ) =

1 2π i

I

∪ j ∂Vj

Q(σ 1/n − qn ) fn h1 dz + z−λ

Z

∪jγj

Q(σ 1/n − qn) fn h ∗ dω z−λ

and so, via (10.3.32), I 1/n − q ) f h 1 Q( σ n n 1 dist(λ , ∂ V j ∩ ∂ ∆ j ) ℓ(λ , σ 1/n fn ) − dz (10.3.33) 2π i (∪ j ∂ V j )∩(∪ j ∆ j ) z−λ

goes to zero as n → ∞ uniformly on ∂ ∆ j ∩(G\V − ) for all j. Now use (10.3.28), (10.3.29), and (10.3.33) to get I 1 Q(σ 1/n − qn ) fn h1 dist(λ , ∂ V j ∩ ∂ ∆ j ) ℓ(λ , g) − dz 2π i (∪ j ∂ V j )∩(∪ j ∆ j ) z−λ goes to zero as n → ∞ uniformly on ∂ ∆ j ∩ (G \ V − ) for all j. In a similar way,

dist(λ , ∂ V j ∩ ∂ ∆ j ) I 1 Q(σ 1/n − qn) fn h1 × ℓ(λ , g) + Q(λ )g(λ )h1(λ ) − dz 2π i (∪ j ∂ V j )∩(∪ j ∆ j ) z−λ

goes to zero as n → ∞ uniformly on ∂ ∆ j ∩V − for all j. Let I 1 Q(σ 1/n − qn) fn h1 un (λ ) := dz 2π i (∪ j ∂ V j )∩(∪ j ∆ j ) z−λ and notice that

b \ ∪ j ∆− ) un ∈ H p (C j

10.4. Finally the proof

111

b \ ∪ j ∆− -outer function with for all 0 < p < 11 . If F is the C j

|F| = dist(λ , ∂ V j ∩ ∂ ∆ j ) on ∂ ∆ j ,

then our work in the previous paragraph shows that the functions (Fun )n>1 form a Cauchy b \ ∪ j ∆− ). then sequence in H 2 (C j Fun → Fℓ(·, g) almost everywhere on (∪ j ∂ ∆ j ) ∩ (G \ V − );

Fun → Fℓ(·, g) + FQgh1 almost everywhere on (∪ j ∂ ∆ j ) ∩V ). By Hardy space theory, Fℓ(·, g)|(∪ j ∂ ∆ j ) ∩ (G \ V − ) b \ ∪ j ∆− ) and so it follows that is part of the boundary function for a function in H 2 (C j − b \ ∪ j ∆ . But, by adjusting the disks ∆ j , we can ℓ(·, g)|G has an analytic continuation to C j

b (the Riemann sphere!) and thus must show that ℓ(·, g)|G has an analytic continuation to C be the constant function. From (10.3.16) we see that ℓ(·, g)|G also has a pseudocontinuation ℓ(·, g)|De . But by uniqueness of pseudocontinuations (Privalov’s uniqueness theorem), the pseudocontinuation and the analytic continuation must be the same. Hence ℓ(·, g), originally defined b which is constant. However, ℓ(∞, g) = 0 and on De ∪ G, has an analytic continuation to C so ℓ(·, g) ≡ 0. In particular, ℓ(·, g) ≡ 0 on De which implies (10.3.9).

10.4 Finally the proof With the technicalities out of the way, we are now ready for the proof of the main result of this chapter. Proof of Theorem 10.1.2. Without loss of generality, we assume that the greatest common G-inner divisor of M is one. Recall that V j are the open sets guaranteed by Lemma 10.2.1 and V=

n [

(Vi ∩ G).

j=1

For f ∈ M, note that k f |V k2H 2 (V ) =

N

∑ k f |V j ∩ Gk2H 2 (V j ∩G)

j=1

− 1 Thus far, we have been talking about Hardy spaces of simply connected domains in C. b The domain C\∪ b j∆ j is not simply connected. However, the known and expected results for the Hardy spaces of simply connected domains still hold for these types of Hardy spaces. We refer the reader to [32, 48, 65, 76] for a discussion of this.


112 and so if

MV := closH 2 (V ) M|V, then M j := (MV )|(V j ∩ G)

is a closed invariant subspace of H 2 (V j ∩ G) for each j = 1, · · · , N. Since each V j is a Carathéodory domain, β j−1 : D → V j is a weak-∗ generator for H ∞ (D) [67]. This means that the polynomials are weak-∗ sequentially dense in H ∞ (V j ). It follows that β j · M j ⊂ M j and so M j ◦ β j−1 is an invariant subspace of H 2 (D\ [0, 1)). (Recall that β j (γ j \ {λ j }) = [0, 1).) Applying Theorem 6.2.1 we obtain E j ⊂ γ j , ρ j : E j → C, and a V j ∩ G-outer function Fj such that g 2 + 2 − M j = g ∈ H (V j ∩ G) : V j ∩ G ∈ H (V j ∩ G), g = ρ j g a.e. on E j . Fj Now let

E :=

N [

E j,

j=1

ρ : E → C, F : V → C,

ρ |E j := ρ j , F|V j ∩ G := Fj .

The result now follows from Proposition 10.3.1.

Chapter 11

Final thoughts Hardy-Smirnov class: It turns out that our results about the invariant subspaces of H 2 (G) can be used to prove analogous results for the Hardy-Smirnov class E 2 (G) (recall the definition from (2.1.9)). We restrict our discussion to the case of the slit disk G = D\[0, 1). If φG is the conformal map from D onto G and ψ = φG−1 , it follows from (2.1.10) that the operator U : H 2 (G) → E 2 (G), U f = (ψ ′ )1/2 f is unitary and moreover, U intertwines S (S f = z f ) on E 2 (G) with S on H 2 (G). Letting K = (ψ ′ )1/2 , we see from the form of ψ ′ (which can be computed from the appendix) that K is a G-outer function. Furthermore, KH 2 (G) = E 2 (G). Thus if M is an invariant subspace of E 2 (G), then K1 M is an invariant subspace of 2 H (G) and so from Theorem 6.2.1, there is a E ⊂ [0, 1], a ρ : E → C, and, for every ε > 0, a Gε -outer function Fε such that f M = KΘM · f ∈ H 2 (G) : ∈ H 2 (Gε ), f + = ρ f − a.e. on E . Fε Setting

ρe :=

K+ ρ K−

and Feε = KFε ,

we see that Feε is Gε -outer and g 2 2 + − M = ΘM · g ∈ E (G) : ∈ H (Gε ), g = ρeg a.e. on E . Feε

(11.0.1)

Banach space case: The main techniques used to characterize the invariant subspaces b \ γ ). It of H 2 (G) depend on our discussion of the nearly invariant subspaces of H 2 (C is here where we used, in key places such as Proposition 3.4.2 and Proposition 3.4.8, Hilbert space techniques such as the Wold decomposition. These do not have Banach b \ γ ) for general p ∈ space analogs and so our techniques do not seem to work for H p (C

Chapter 11. Final thoughts

114

[1, ∞). The main results of this monograph (Theorem 3.1.2 and Theorem 6.2.1) should be true in the H p setting but will require Banach space techniques. Geometry: Our results characterize the invariant subspaces of H 2 (G) for a slit domain where the analytic slit meets the boundary of ∂ G at a positive angle. Does anything change when the slit is tangent to ∂ G? Can we relax the condition that the slit is analytic? Does the slit domain need to be simply connected? What happens when there are a countably infinite number of slits? Cyclic invariant subspaces: For certain special f ∈ H 2 (G) (Theorem 7.2.2 and Corollary 7.2.6), we can compute [ f ], the invariant subspace generated by f . What is [ f ] for a general f ∈ H 2 (G)? We also know (Remark 7.2.1) that not every invariant subspace is cyclic. Which ones are? For example, from Corollary 8.2.11 we see that if M is cyclic then σe (S|M) = T. Is the converse true? Lattice operations: For any bounded domain Ω, the invariant subspaces of H 2 (Ω) form a lattice in that if M1 and M2 are invariant subspaces, then so are M1 ∨ M2 and M1 ∩ M2 . When Ω = D, we know from Beurling’s theorem that M1 = Θ1 H 2 (D) and M2 = Θ2 H 2 (D) for some D-inner functions Θ1 and Θ2 . Moreover [23, p. 137], M1 ∨ M2 = g.c.d.(Θ1 , Θ2 )H 2 (D),

M1 ∩ M2 = l.c.m.(Θ1 , Θ2 )H 2 (D).

When Ω = G \ [0, 1), Theorem 6.2.1 says that the invariant subspaces M1 and M2 depend on the parameters Θ j , ρ j , E j , Fj , j = 1, 2. Can one describe M1 ∩ M2 and M1 ∨ M2 in terms of these parameters?

Chapter 12

Appendix In this appendix we store some information about two particular conformal maps used in this monograph. We first mention a few words about the conformal map

φG : D → D \ [0, 1). Consider the following sequence of conformal maps w1 (z) := i w2 (z) = w3 (z) :=

1+z : D → {ℑz > 0}; 1−z

√ z : {ℑz > 0} → {ℑz > 0} ∩ {ℜz > 0},

z−1 : {ℑz > 0} ∩ {ℜz > 0} → D ∩ {ℑz > 0}. z+1

Notice that w1 ({eit : π < t < 3π /2}) = (0, 1), w2 (0, 1) = (0, 1), w3 (0, 1) = (−1, 0), Define

w1 ({eit : 3π /2 < t < 2π }) = (1, ∞); w2 (1, ∞) = (1, ∞); w3 (1, ∞) = (0, 1).

φG (z) = (w3 (w2 (w1 (−iz)))2

and notice that φG is a conformal map from D onto the slit disk G = D \ [0, 1) and, by following the boundaries, we see that φG maps the arc {eit : 0 < t < π /2} to the top half of the slit while φG maps {e−it : 0 < t < π /2} to the bottom half of the slit. Furthermore, it is routine to show that φG maps the interval (−1, 1) onto (−1, 0) and so, by the Schwarz reflection principle, φG has the following nice property

φG (eiθ ) = φG (e−iθ ),

0 6 θ 6 2π .

Chapter 12. Appendix

116 The previous identity says that if ψG = φG−1 , then

ψG+ (x) = ψG− (x), and so

0 < x < 1,

|(ψG′ )+ (x)| = |(ψG′ )− (x)|,

0 < x < 1.

Next, we say a few words about the conformal map b \ γ, α : D \ [0, 1) → C

γ = {eit : −π /2 6 t 6 π }.

We construct this map with a sequence of standard conformal maps: u1 (z) =

1+z : D \ [0, 1) → {ℜz > 0} \ [1, ∞); 1−z

b \ ((−∞, 0] ∪ [1, ∞)), u2 (z) = z2 : {ℜz > 0} \ [1, ∞) → C u3 (z) :=

Observe that

z−i b b \ γ, : C \ ((−∞, 0] ∪ [1, ∞)) → C z+i u1 (0, 1) = (1, ∞),

u1 (D+ ) = {ℜz > 0} ∩ {ℑz > 0}, u2 ({ℜz > 0} ∩ {ℑz > 0}) = {ℑz > 0}, u3 (−∞, 0] = γ , ′

u3 [1, ∞) = γ ′′ ,

and note that

u2 ({ℜz > 0} ∩ {ℑz < 0}) = {ℑz < 0};

γ := {eit : 0 6 t 6 π }, ′

γ ′′ := {eit : −π /2 6 t 6 0};

u3 ({ℑz > 0}) = D, From here, we define

u1 (D− ) = {ℜz > 0} ∩ {ℑz < 0};

b \ D− . u3 ({ℑz < 0}) = C

α := u3 ◦ u2 ◦ u1

b \ γ, α : D \ [0, 1) → C

α [0, 1] = γ ′′ ,

α (T ∩ {ℑz > 0}) = γ ′ , α (D+ ) = D,

α (T ∩ {ℑz < 0}) = γ ′ ,

b \ D− . α (D− ) = C

Furthermore, it is easy to check from the identity

α (z) = that

(1 + z)2 (1 − z)−2 − i (1 + z)2 (1 − z)−2 + i

α (e−iθ ) = α (eiθ ).

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Index Ω-inner, 20 Ω-outer, 20

harmonic measure, 12, 14, 16, 18, 19, 69, 101

backward shift operator, 6, 47, 49 Bergman space, v Beurling’s theorem, xvii, 1, 21 Blaschke product, xv bounded point evaluations, 7, 10, 83 bounded type, xvi

inner function, 1, 20 invariant subspace, 1, 65, 66, 68, 97, 113

Cauchy integral formula, xiv, 23 Cauchy transform, 34, 47, 51 closure of polynomials, 7, 82 compression, 94 corner, 11, 69 cyclic, 6, 80–82, 89, 90, 114 de Branges-type space, 6, 59, 62 Dirichlet problem, 9, 12 Dirichlet space, v essential spectrum, 7, 85, 87, 89, 90 extremal function, 27 F. and M. Riesz theorem, xv, 40 factorization, xv, 20 Fatou’s jump theorem, 35 Fredholm theory, 85 G-inner, 3 G-outer, 4 Hardy space, 9, 22 Hardy-Smirnov class, 11, 22, 113 harmonic majorant, 9

kernel function, 10, 27, 29, 61 least harmonic majorant, 9 Morera’s theorem, 46, 69 multiple slit domain, 97 multiplicity, 37 nearly invariant, 5, 25, 47, 49, 56, 62, 66 Nevanlinna class, xvi, 21 non-tangential limit, xvii normalized reproducing kernel, 27, 29, 70, 71 norming point, 9, 48 outer function, xv, 20 parameters, 46, 95 Poisson kernel, 35 Polubarinova and Kochina theorem, xvi, 22 Privalov’s uniqueness theorem, xvii, 41, 73 pseudocontinuation, 47 reproducing kernel, 10 Riesz theorem, 21 singular inner function, xvi slit domain, 1, 14, 18, 97

124 Smirnov class, xvi, 21 Smirnov’s theorem, xv spectrum, 85 Stoltz region, xvi subharmonic, 9 subspace, 1 two-cyclic, 6, 79 unilateral shift, 37, 39 von Neumann-Wold decomposition, 5, 37 Wold decomposition, 5, 37

Index

The Hardy space of a slit domain

The Hardy Space of a Slit Domain (Frontiers in Mathematics Series)

Domain (The Domain Trilogy)

The domain of language

Domain

Domain

Domain

Domain

A Darker Domain

Domain

Domain

Domain

The Flight of Gemma Hardy

A Darker Domain

A Darker Domain

Domain

Domain

Domain

Domain

Domain

A Darker Domain: A Novel

A Darker Domain

A Darker Domain

The Flight of Gemma Hardy

Domain

An Introduction to Operators on the Hardy-Hilbert Space

An Introduction to Operators on the Hardy-Hilbert Space

A Darker Domain

A Darker Domain

A Darker Domain

An Introduction to Operators on the Hardy-Hilbert Space

The Hardy space of a slit domain