Operators on harmonic Bergman spaces

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 theL ^p 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.Partially supported by a grant from the Research Grants Committee of the University of Alabama.     

Positive Toeplitz Operators Between the Harmonic Bergman Spaces

On the setting of the upper half space we study positive Toeplitz operators between the harmonic Bergman spaces. We give characterizations of bounded and compact positive Toeplitz operators taking a harmonic Bergman space b ^p into another b ^q for 1<p<, 1<q<. The case p=1 or q=1 seems more intriguing and is left open for further investigation. Also, we give criteria for positive Toeplitz operators acting on b ² to be in the Schatten classes. Some applications are also included.     

Positive Toeplitz operators between the pluriharmonic Bergman spaces

We study Toeplitz operators between the pluriharmonic Bergman spaces for positive symbols on the ball. We give characterizations of bounded and compact Toeplitz operators taking a pluriharmonic Bergman space b ^p into another b ^q for 1 < p, q < ∞ in terms of certain Carleson and vanishing Carleson measures. This research was supported by KOSEF (R01-2003-000-10243-0) and Korea University Grant.     

Bounded symbols and Reproducing Kernel Thesis for truncated Toeplitz operators

Compressions of Toeplitz operators to coinvariant subspaces of H² are called truncated Toeplitz operators. We study two questions related to these operators. The first, raised by Sarason, is whether boundedness of the operator implies the existence of a bounded symbol; the second is the Reproducing Kernel Thesis. We show that in general the answer to the first question is negative, and we exhibit some classes of spaces for which the answers to both questions are positive.     

Commutativity of kth‐order slant Toeplitz operators

In this paper, commutativity of k^th‐order slant Toeplitz operators are discussed. We show that commutativity and essential commutativity of two slant Toeplitz operators are the same. Also, we study k^th‐order slant Toeplitz operators on the Bergman space L²_a(D) and give some commuting properties, algebraic and spectral properties of k^th‐order slant Toeplitz operators on the Bergman space (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the reflexivity of subspaces of Toeplitz operators on the hardy space on the upper half-plane

The reflexivity and transitivity of subspaces of Toeplitz operators on the Hardy space on the upper half-plane are investigated. The dichotomic behavior (transitive or reflexive) of these subspaces is shown. It refers to the similar dichotomic behavior for subspaces of Toeplitz operators on the Hardy space on the unit disc. The isomorphism between the Hardy spaces on the unit disc and the upper half-plane is used. To keep weak* homeomorphism between L ^∞ spaces on the unit circle and the real line we redefine the classical isomorphism between L ¹ spaces.     

Toeplitz operators on Herglotz wave functions

The space of Herglotz wave functions in R² consists of all the solutions of the Helmholtz equation that can be represented as the Fourier transform in R² of a measure supported in the circle and with density in L²(S¹). This space has a structure of a Hilbert space with reproducing kernel. The purpose of this article is to study Toeplitz operators with nonnegative radial symbols, defined on this space. We study the symbols defining bounded and compact Toeplitz operators as well as the Toeplitz operators belonging to the Schatten classes sp.     

Toeplitz operators on generalised H2 spaces

Some aspects are developed of the theory of Toeplitz operators on generalised HardyH ² spaces associated to function algebras. It is shown that a substantial number of results of the classical theory of Toeplitz operators on the circle extend to this situation, although counterexamples are given which show that there are also important differences. Spectral connectedness results are obtained, and a characterisation of invertibility for Toeplitz operators. The Fredholm theory is also studied.     

Maximal Abelian von Neumann Algebras and Toeplitz Operators with Separately Radial Symbols

This paper mainly concerns abelian von Neumann algebras generated by Toeplitz operators on weighted Bergman spaces. Recently a family of abelian w^*-closed Toeplitz algebras has been obtained (see [5,6,7,8]). We show that this algebra is maximal abelian and is singly generated by a Toeplitz operator with a “common” symbol. A characterization for Toeplitz operators with radial symbols is obtained and generalized to the high dimensional case. We give several examples for abelian von Neumann algebras in the case of high dimensional weighted Bergman spaces, which are different from the one dimensional case.     

Commutative Toeplitz Banach Algebras on the Ball and Quasi-Nilpotent Group Action

Studying commutative C*-algebras generated by Toeplitz operators on the unit ball it was proved that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C*-algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each standard weighted Bergman space. There are five different pairwise non-conjugate model classes of such subgroups: quasi-elliptic, quasi-parabolic, quasi-hyperbolic, nilpotent and quasi-nilpotent. Recently it was observed in Vasilevski (Integr Equ Oper Theory. 66:141–152, 2010) that there are many other, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were subordinated to the quasi-elliptic group, the corresponding commutative operator algebras were Banach, and being extended to C*-algebras they became non-commutative. These results were extended then to the classes of symbols, subordinated to the quasi-hyperbolic and quasi-parabolic groups. In this paper we prove the analogous commutativity result for Toeplitz operators whose symbols are subordinated to the quasi-nilpotent group. At the same time we conjecture that apart from the known C*-algebra cases there are no more new Banach algebras generated by Toeplitz operators whose symbols are subordinated to the nilpotent group and which are commutative on each weighted Bergman space.     

重调和Hardy空间上的Toeplitz 算子

与调和Bergman 空间相对应, 本文研究重调和Hardy 空间h²(D²) 上的Toeplitz 算子. 本文发现, h²(D²) 上的Toeplitz 算子与经典的Hardy 空间、Bergman 空间及调和Bergman 空间上的Toeplitz算子的性质都有很大的差异. 例如解析Toeplitz 算子可以不是半可换及可交换的. 即使半可换, 其中任何一个符号可以不为常数; 即使可交换, 两个符号的非平凡线性组合也不一定是常数. 本文得到了h²(D²) 上两个解析Toeplitz 算子半可换和可换的充分必要条件.     

Toeplitz Operators on Generalized Bergman Spaces

We consider the weighted Bergman spaces HL²(\mathbb B^d, m_l){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dm_l(z) = c_l(1-|z|²)^l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.     

Toeplitz Operators and Division Theorems in Anisotropic Spaces of Holomorphic Functions in the Polydisc

This work is an introduction to anisotropic spaces, which have an ω-weight of analytic functions and are generalizations of Lipshitz classes in the polydisc. We prove that these classes form an algebra and are invariant with respect to monomial multiplication. These classes are described in terms of derivatives. It is established that Toeplitz operators are bounded in these (Lipschitz and Djrbashian) spaces. As an application, a theorem about the division by good-inner functions in the mentioned classes is proved.     

Toeplitz Operators on the Fock Space

We study Toeplitz operators on the Fock space with positive measures as symbols. Main results include characterizations of Fock–Carleson measures, bounded Toeplitz operators, compact Toeplitz operators, and Toeplitz operators in the Schatten p-classes.     

Unbounded Toeplitz Operators

This partly expository article develops the basic theory of unbounded Toeplitz operators on the Hardy space H ², with emphasis on operators whose symbols are not square integrable. Unbounded truncated Toeplitz operators on coinvariant subspaces of H ² are also studied. In memory of Paul R. Halmos     

Toeplitz Operators on Arveson and Dirichlet Spaces

We define Toeplitz operators on all Dirichlet spaces on the unit ball of and develop their basic properties. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols in terms of Carleson measures and Berezin transforms. Our results naturally extend those known for weighted Bergman spaces, a special case applies to the Arveson space, and we recover the classical Hardy-space Toeplitz operators in a limiting case; thus we unify the theory of Toeplitz operators on all these spaces. We apply our operators to a characterization of bounded, compact, and Schatten-class weighted composition operators on weighted Bergman spaces of the ball. We lastly investigate some connections between Toeplitz and shift operators. The research of the second author is partially supported by a Fulbright grant.     

Toeplitz operators on harmonic Bergman spaces

We study Toeplitz operators on the harmonic Bergman spaceb ^p (B), whereB is the open unit ball inR ⁿ(n2), for 1<p. We give characterizations for the Toeplitz operators with positive symbols to be bounded, compact, and in Schatten classes. We also obtain a compactness criteria for the Toeplitz operators with continuous symbols.     

Some results for operators on a model space

We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space K _θ = H²ΘθH² is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated Toeplitz operators. Moreover, we study some property for H²-functions in terms of noncyclicity of co-analytic Toeplitz operators and hypercyclicity of model operators.     

Properties of representations of operators acting between spaces of vector-valued functions

A well-known result going back to the 1930s states that all bounded linear operators mapping scalar-valued L ¹-spaces into L ^∞-spaces are kernel operators and that in fact this relation induces an isometric isomorphism between the space of such operators and the space of all bounded kernels. We extend this result to the case of spaces of vector-valued functions. A recent result due to Arendt and Thomaschewski states that the local operators acting on L ^p -spaces of functions with values in separable Banach spaces are precisely the multiplication operators. We extend this result to non-separable dual spaces. Moreover, we relate positivity and other order properties of the operators to corresponding properties of the representations.     

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.