Theory of Bergman Spaces

In both the Bergman space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A squared"> <mml:semantics> <mml:msup> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">A^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Hardy space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H squared"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the problem of determining which bounded univalent mappings of the unit disk have the wandering property is addressed. Generally, a function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">H^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has the wandering property in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A squared"> <mml:semantics> <mml:msup> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">A^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H squared"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , provided that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant subspace <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is generated by the orthocomplement of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g upper M"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">gM</mml:annotation> </mml:semantics> </mml:math> </inline-formula> within <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It is known that essentially every function which has the wandering property in either space is the composition of a univalent mapping with a classical inner function, and that the identity mapping has this property in both spaces. Consequently, weak-star generators of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">H^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> also have the wandering property in both settings. The present paper gives a partial converse to this, and shows that in both settings there is a large class of bounded univalent mappings which fail to have the wandering property.

Products of Hankel and Toeplitz Operators on the Bergman Space

In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity.

In this paper one considers some general theorems of the theory of Hankel and Toeplitz operators in spaces of analytic functions. Under natural restrictions on the spaces X, it is shown that the symbols of the Toeplitz operators, acting in X, are bounded. One describes completely the symbols of the Hankel and Toeplitz operators, acting from Hp into ¯Hq (into Hq) for 0 < p, q < ∞.

Let L2 = L2(D, r dr dθ/π) be the Lebesgue space on the open unit disc and let be the Bergman space. Let P be the orthogonal projection of L2 onto and let Q be the orthogonal projection onto . Then I − P ≥ Q. The big Hankel operator and the small Hankel operator on are defined as: for ϕ in L∞, and . In this paper, the finite‐rank intermediate Hankel operators between and are studied. We are working on the more general space, that is, the weighted Bergman space.

This paper is a survey of some works on paracommutators and Hankel operators in China, which are the subjects I have been studying since 1985. The classical Hankel operator is defined on the Hilbert space l 2 by the Hankel matrix Γ = {b n+k } n,k ≥0, and the Toeplitz operator is defined also on l 2 by the toeplitz matrix T = {b n−k } n≥k≥0. By the theory of the Fourier series both of them become the operators on the Hardy space \( {{H}^{2}}(T) = \{ f \in {{L}^{2}}(T):\hat{f}(n) = 0,n < 0\} . \). Let \( \overline H ^2 (T) \) denote the conjugate Hardy space and P,P denote the ortnogonal projections from L 2(T) into H 2(T),H 2 (T) respectively. Then the Toeplitz operator is defined as follows $$ {T_b}(f) = P(\bar bf),for f \in {H^2}(T) $$ (1.1) and the Hankel operator is defined as follows $$ {H_b}(f) = \bar P(\bar bf),for f \in {H^2}(T).$$ (1.2)

One of the major goals in the theory of Toeplitz operators on the Bergman space over the unit disk D in the complex place C is to completely describe the commutant of a given Toeplitz operator, that is, the set of all Toeplitz operators that commute with it. Here we shall study the commutants of a certain class of quasihomogeneous Toeplitz operators defined on the harmonic Bergman space.

For 1le p<infty , let A^{p}_{omega } be the weighted Bergman space associated with an exponential type weight ω satisfying \t\t\t∫D|Kz(ξ)|ω(ξ)1/2dA(ξ)≤Cω(z)−1/2,z∈D,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\int _{{\\mathbb{D}}} \\bigl\\vert K_{z}(\\xi ) \\bigr\\vert \\omega (\\xi )^{1/2} \\,dA(\\xi ) \\le C \\omega (z)^{-1/2}, \\quad z\\in {\\mathbb{D}}, $$\\end{document} where K_{z} is the reproducing kernel of A^{2}_{omega }. This condition allows us to obtain some interesting reproducing kernel estimates and more estimates on the solutions of the ∂̅-equation (Theorem 2.5) for more general weight omega _{*}. As an application, we prove the boundedness of the Bergman projection on L^{p}_{omega }, identify the dual space of A^{p}_{omega }, and establish an atomic decomposition for it. Further, we give necessary and sufficient conditions for the boundedness and compactness of some operators acting from A^{p}_{omega } into A^{q}_{omega }, 1le p,q<infty , such as Toeplitz and (big) Hankel operators.

We derive an asymptotic formula for the argument of a Blaschke product in the upper half-plane with purely imaginary zeros. We then use this formula to find conditions for the quotient of two such Blaschke products to be continuous on the real line. These results are applied to certain Hankel and Toeplitz operators arising in the Cauchy problem for the Korteweg-de Vries equation. Our main theorems include certain compactness conditions for Hankel operators and invertibility conditions for Toeplitz operators with oscillating symbols containing such quotients. As a by-product we obtain a well-posedness result for the Korteweg-de Vries equation.

We study the commutator of the multiplication and harmonic Bergman projection, Hankel and Toeplitz operators on the harmonic Bergman spaces. The same type operators have been well studied on the analytic Bergman spaces. The main difficulty of this study is that the bounded harmonic function space is not an algebra! In this paper, we characterize theLp boundedness and compactness of these operators with harmonic symbols. Results about operators in Schatten classes, the cut-off phenomenon and general symbols are also included.

An equation with a Hardy space Toeplitz operator can be solved by Wiener–Hopf factorization. However, Wiener–Hopf factorization does not work for Bergman space Toeplitz operators. The only way we see to tackle equations with a Toeplitz operator on the Bergman space is to have recourse to approximation methods. The paper is intended as a review of and an illustrated tour through several such methods and thus through some beautiful topics in the intersection of operator theory, complex analysis, differential geometry, and numerical mathematics.

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, the C&lt;sup&gt;*&lt;/sup&gt;-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spaces A&lt;sup&gt;2&lt;/sup&gt;(Ω) for a wide class of plane domains Ω⊂C, and in Fock spaces A&lt;sup&gt;2&lt;/sup&gt;(C&lt;sup&gt;N&lt;/sup&gt;), N≧1.

The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces $$A_{\alpha }^p$$ with $$p \in (0, \infty )$$ and $$ \alpha \in (-1, \infty )$$ . In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

The paper deals with the invertibility of Toeplitz plus Hankel operators T(a)+H(b) acting on classical Hardy spaces on the unit circle T. It is supposed that the generating functions a and b satisfy the condition a(t)a(1/t) = b(t)b(1/t), t ∈ T. Special attention is paid to the case of piecewise continuous generating functions. In some cases the dimensions of null spaces of the operator T(a) + H(b) and its adjoint are described. Fredholm properties of Toeplitz plus Hankel operatorsT(a)+H(b) with piecewise continuous generating functions a and b have been studied for many years. These operators are considered in various Banach and Hilbert spaces, and the results ob- tained show that the structure of the algebras generated by such operators is much more complicated than the structure of the algebras generated by one-dimensional singular integral operators with piecewise continuous coefficients defined on closed smooth curves. Moreover, calculating the index of Toeplitz plus Hankel operators, one encounters even more difficult problems, and the difficulti es grow if one attempts to study invertibility or one-sided invertibility of such operators. It is worth noting that, in general, Fredholm operators T(a) +H(b) are not one-sided invertible. Nev- ertheless, a few works where one-sided invertibility was discussed, have appeared in literature recently. They are mainly concerned with two special cases of Toeplitz plus Hankel operators—viz. with the operators having the form M(a) := T(a) +H(a) or the form f

Let D be a bounded strongly pseudoconvex domain with smooth boundary in Cn and let f∈L2(D). For the Hankel operator Hf on the Bergman space A2(D), it is shown that the essential norm of Hf in L2(D) is comparable to the distance norm from Hf to compact Hankel operators. The result extends the previous corresponding version in the disc proved by Lin and Rochberg in Integ.Equat.Oper.Theory 361–372,17 (1993).