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## Über dieses Buch

Preliminary Text. Do not use. 15 years ago the function theory and operator theory connected with the Hardy spaces was well understood (zeros; factorization; interpolation; invariant subspaces; Toeplitz and Hankel operators, etc.). None of the techniques that led to all the information about Hardy spaces worked on their close relatives the Bergman spaces. Most mathematicians who worked in the intersection of function theory and operator theory thought that progress on the Bergman spaces was unlikely. Now the situation has completely changed. Today there are rich theories describing the Bergman spaces and their operators. Research interest and research activity in the area has been high for several years. A book is badly needed on Bergman spaces and the three authors are the right people to write it.

## Inhaltsverzeichnis

### 1. The Bergman Spaces

Abstract
In this chapter we introduce the Bergman spaces and concentrate on the general aspects of these spaces. Most results are concerned with the Banach (or metric) space structure of Bergman spaces. Almost all results are related to the Bergman kernel. The Bloch space appears as the image of the bounded functions under the Bergman projection, but it also plays the role of the dual space of the Bergman spaces for small exponents (0 < p ≤ 1).
Haakan Hedenmalm, Boris Korenblum, Kehe Zhu

### 2. The Berezin Transform

Abstract
In this chapter we consider an analogue of the Poisson transform in the context of Bergman spaces, called the Berezin transform. We show that its fixed points are precisely the harmonic functions. We introduce a space of BMO type on the disk, the analytic part of which is the Bloch space, and characterize this space in terms of the Berezin transform.
Haakan Hedenmalm, Boris Korenblum, Kehe Zhu

### 3. A p -Inner Functions

Abstract
In this chapter, we introduce the notion of A p α -inner functions and prove a growth estimate for them. The A p α -inner functions are analogous to the classical inner functions which play an important role in the factorization theory of the Hardy spaces. Each A p α -inner function is extremal for a z-invariant subspace, and the ones that arise from subspaces given by finitely many zeros are called finite zero extremal functions (for α = 0, they are also called finite zero-divisors). In the unweighted case α = 0, we will prove the expansive multiplier property of A up -inner functions, and obtain an “inner-outer”-type factorization of functions in A p . In the process, we find that all singly generated invariant subspaces are generated by its extremal function. In the special case of p = 2 and α = 0, we find an analogue of the classical Carathéodory-Schur theorem: the closure of the finite zero-divisors in the topology of uniform convergence on compact subsets are the A2-subinner functions. In particular, all A2-inner functions are norm approximable by finite zero-divisors.
Haakan Hedenmalm, Boris Korenblum, Kehe Zhu

### 4. Zero Sets

Abstract
For an analytic function f in D, not identically zero, we let Zf denote the zero sequence of f , with multiple zeros repeated according to multiplicities. A sequence A = {a n } n in D is called a zero set for Apα if there exists a nonzero function fA p α such that A = Z f , counting multiplicities. Zero sets for other spaces of analytic functions are defined similarly.
Haakan Hedenmalm, Boris Korenblum, Kehe Zhu

### 5. Interpolation and Sampling

Abstract
In this chapter, we define and study sequences of interpolation and sampling for the Bergman spaces A and Ap α . The main results include the characterization of interpolation sequences in terms of an upper density and the characterization of sampling sequences in terms of a lower density.
Haakan Hedenmalm, Boris Korenblum, Kehe Zhu

### 6. Invariant Subspaces

Abstract
In this chapter we study several problems related to invariant subspaces of Bergman spaces. First, we show by explicit examples that there exist invariant subspaces of index n for all 0 ≤ n ≤ +∞. Then we prove a theorem that can be considered an analogue to the classical Beurling’s theorem on invariant subspaces of the Hardy space. It states that in the spaces A2 α, with — 1 < α ≤ 0, each invariant subspace I is generated by I ⊖z I. In the classical Hardy space case, IzI is one-dimensional, and spanned by a classical inner function (unless I = {0}). In A2α, the dimension may be bigger, but all elements of Izl of unit norm are A2α-inner functions
Haakan Hedenmalm, Boris Korenblum, Kehe Zhu

### 7. Cyclicity

Abstract
In this chapter, we study the cyclic functions in the Bergman spaces Ap. First, we identify them with the Ap-outer functions, which are defined in terms of a notion of domination, in a fashion analogous to what is done in the classical Hardy space setting. Second, we show that a function that belongs to a smaller space Aq, p < q, is cyclic in Ap if and only if it is cyclic in the growth space A-∞. Then we characterize the cyclic vectors for. A-∞ in terms of boundary premeasures; this constitutes the bulk of the material in the chapter
Haakan Hedenmalm, Boris Korenblum, Kehe Zhu

### 8. Invertible Noncyclic Functions

Abstract
A function f in a space X of analytic functions is said to be invertible if 1/f also belongs to X. In the classical theory of Hardy spaces, every invertible function in Hp is necessarily cyclic in Hp. This is also true in the A-∞ theory; an invertible function in A-∞ is always cyclic in A-∞.
Haakan Hedenmalm, Boris Korenblum, Kehe Zhu

### 9. Logarithmically Subharmonic Weights

Abstract
In this chapter, we study weighted Bergman spaces for weights that are logarithmically subharmonic and reproduce for the origin; the latter means that if we integrate a bounded harmonic function against the weight over D, we obtain the value of the harmonic function at the origin. Two important examples of such weights are ω(z) = |Gz)| p , where G is an Ap-inner function, and ω(z) = (α + 1)(1 — |z|2)α,z ∈ D,where —1 < α ≤ 0. Not only are these weights interesting in themselves, they also have nice applications to the study of unweighted Bergman spaces. The main result of the chapter is that the weighted biharmonic Green function гω is positive, provided that the weight is logarithmically subharmonic and reproduces for the origin. As a consequence, we will prove the domination relation ‖GAfp ≤ ‖G Bfp, where f is any function in Ap, and GAand GB are contractive zero divisors in ApwithA ⊂ B.
Haakan Hedenmalm, Boris Korenblum, Kehe Zhu

### Backmatter

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