Norms of Toeplitz and Hankel Operators on Hardy Type Subspaces of Rearrangement-Invariant Spaces

We prove analogues of the Brown-Halmos and Nehari theorems on the norms of Toeplitz and Hankel operators, respectively, acting on subspaces of Hardy type of reflexive rearrangement-invariant spaces with nontrivial Boyd indices.     

Toeplitz Operators and Herz Spaces on the Half-Space

On the setting of the half-space we introduce the Schatten-Herz classes of Toeplitz operators and obtain characterizations for positive Toeplitz operators to belong to those classes. We also prove results concerning the boundedness and compactness of Toeplitz operators with Herz symbols. Such a study has been recently done on the ball. At a critical step of the proofs we employ a much simplified argument to extend the range of parameters for Herz spaces on which the Berezin transform is bounded. Our results show not only that most of results on the ball continue to hold, but also that there is some pathology caused by the unboundedness of the domain. The first author was in part supported by a Korea University Grant(2007), the second author was in part supported by Hanshin University Research Grant, and both authors were in part supported by KOSEF(R01-2003-000-10243-0).     

Norms of Moore-Penrose Inverses of Fredholm Toeplitz Operators

In this paper we estimate the norm of the Moore-Penrose inverse T(a)⁺ of a Fredholm Toeplitz operator T(a) with a matrix-valued symbol a∈L_{N × N}^∞ defined on the complex unit circle. In particular, we show that in the ”generic case” the strict inequality ||T(a)⁺|| > ||a⁻¹||_∞ holds. Moreover, we discuss the asymptotic behavior of ||T(t^ra)⁺|| for . The results are illustrated by numerical experiments.     

Dynamics of Properties of Toeplitz Operators with Radial Symbols

In the case of radial symbols we study the behavior of different properties (boundedness, compactness, spectral properties, etc.) of Toeplitz operators T_a⁽⁾ acting on weighted Bergman spaces over the unit disk , in dependence on , and compare their limit behavior under with corresponding properties of the initial symbol a.     

A Note on the Range of Toeplitz Operators

A well known lemma attributed to Coburn states that a Toeplitz operator with non-trivial kernel acting on the Hardy space must have dense range. We show that the range of a non-zero Toeplitz operator with non-trivial kernel must contain all polynomials and state this in a precise form.     

The Reproducing Kernel Thesis for Toeplitz Operators on the Paley-Wiener Space

It is known that for particular classes of operators on certain reproducing kernel Hilbert spaces, key properties of the operators (such as boundedness or compactness) may be determined by the behaviour of the operators on the reproducing kernels. We prove such results for Toeplitz operators on the Paley-Wiener space, a reproducing kernel Hilbert space over . Namely, we show that the norm of such an operator is equivalent to the supremum of the norms of the images of the normalised reproducing kernels of the space. In particular, therefore, the operator is bounded exactly when this supremum is finite. In addition, a counterexample is provided which shows that the operator norm is not equivalent to the supremum of the norms of the images of the real normalised reproducing kernels. We also give a necessary and sufficient condition for compactness of the operators, in terms of their limiting behaviour on the reproducing kernels.     

Generalization of the Newman-Shapiro isometry theorem and Toeplitz operators

We give a generalization of the Newman-Shapiro Isometry Theorem to the case of Hilbert space-valued entire functions, which are square-summable with respect to the Gaussian measure on ⁿ , together with some applications in the theory of Toeplitz operators with operator-valued symbols. The study of various properties (such as density of domains, cores, closedness and boundedness from below) of these operators in illustrated with many relevant examples.Research supported by KBN under grant no. 2 P03A 041 10.     

Products of Toeplitz Operators on the Bergman Space

In 1962 Brown and Halmos gave simple conditions for the product of two Toeplitz operators on Hardy space to be equal to a Toeplitz operator. Recently, Ahern and Cucković showed that a similar result holds for Toeplitz operators with bounded harmonic symbols on Bergman space. For general symbols, the situation is much more complicated. We give necessary and sufficient conditions for the product to be a Toeplitz operator (Theorem 6.1), an explicit formula for the symbol of the product in certain cases (Theorem 6.4), and then show that almost anything can happen (Theorem 6.7).     

Compact Operators on Bergman Spaces

We prove that a bounded operator S on L _a ^p for p > 1 is compact if and only if the Berezin transform of S vanishes on the boundary of the unit disk if S satisfies some integrable conditions. Some estimates about the norm and essential norm of Toeplitz operators with symbols in BT are obtained.     

Toeplitz Operators on Analytic Besov Spaces

We characterize complex measures μ on the unit disk for which the Toeplitz operator T _μ is bounded or compact on the analytic Besov spaces B _p with 1 ≤ p < ∞. Research supported in part by NSF grant, DMS 0200587 (first author); and by a NSERC grant (third author).     

Composition Operators on Orlicz–Lorentz Spaces

In this paper, we study the boundedness and the compactness of composition operators on Orlicz–Lorentz spaces.      

Finite Interval Convolution Operators with Transmission Property

We study convolution operators in Bessel potential spaces and (fractional) Sobolev spaces over a finite interval. The main purpose of the investigation is to find conditions on the convolution kernel or on a Fourier symbol of these operators under which the solutions inherit higher regularity from the data. We provide conditions which ensure the transmission property for the finite interval convolution operators between Bessel potential spaces and Sobolev spaces. These conditions lead to smoothness preserving properties of operators defined in the above-mentioned spaces where the kernel, cokernel and, therefore, indices do not depend on the order of differentiability. In the case of invertibility of the finite interval convolution operator, a representation of its inverse is presented in terms of the canonical factorization of a related Fourier symbol matrix function.     

Finite Rank Perturbations of Toeplitz Operators

We study finite rank perturbations of the Brown-Halmos type results involving products of Toeplitz operators acting on the Bergman space.      

Composition Operators and Vector-valued BMOA

Analytic composition operators are studied on X-valued versions of BMOA, the space of analytic functions on the unit disk that have bounded mean oscillation on the unit circle, where X is a complex Banach space. It is shown that if X is reflexive and C _φ is compact on BMOA, then C _φ is weakly compact on the X-valued space BMOA _C (X) defined in terms of Carleson measures. A related function-theoretic characterization is given of the compact composition operators on BMOA.     

Automorphisms on the Toeplitz Algebra with Piecewise Continuous Symbol on the Unit Ball

In this paper, we obtain a Fredholm index formula for Toeplitz operators whose symbols are certain piecewise continuous function matrices on the unit ball. Moreover, using this formula, we discuss the automorphisms on the corresponding Toeplitz algebra     

Composition Operators on Small Spaces

We show that if a small holomorphic Sobolev space on the unit disk is not just small but very small, then a trivial necessary condition is also sufficient for a composition operator to be bounded. A similar result for holomorphic Lipschitz spaces is also obtained. These results may be viewed as boundedness analogues of Shapiro’s theorem concerning compact composition operators on small spaces. We also prove the converse of Shapiro’s theorem if the symbol function is already contained in the space under consideration. In the course of the proofs we characterize the bounded composition operators on the Zygmund class. Also, as a by-product of our arguments, we show that small holomorphic Sobolev spaces are algebras.     

On the Essential Commutant of Toeplitz Operators in Several Complex Variables

Using the joint local mean oscillation, Jingbo Xia [13] showed that the essential commutant of , where is the subalgebra of L ^∞ generated by all functions which are bounded and have at most one discontinuity, is (QC). Even though Xia’s method cannot be used, we are able to generalize his result to Toeplitz operators in higher dimensions with a different approach. This result is stronger than the well-known result stating that the essential commutant of the full Toeplitz algebra is (QC).      

Commutants of Analytic Toeplitz Operators on the Harmonic Bergman Space

We study the commuting problem for Toeplitz operators on the harmonic Bergman space of the unit disk. We show that an analytic Toeplitz operator and a co-analytic Toeplitz operator with certain noncyclicity hypothesis can commute only when one of their symbols is constant. We also obtain analogous results for semi-commutants.     

Carleson Type Theorems for Certain Convolution Operators

We prove mapping theorems for some convolution operators, acting from Sobolev type spaces in to Lorentz spaces defined on with a fractional-order Carleson measure. As an application of the major theorems, we give some a priori estimates for the solutions of certain elliptic equations.     

Almost-invertible Toeplitz operators and K-theory

We consider the question of when a Toeplitz operator with continuous symbol on a connected compact abelian group is almost invertible, and show that this occurs precisely when the symbol is invertible and has zero topological index. The proof uses someK-theory computations.