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.     

Hankel and Toeplitz operators on Dirichlet spaces

In this paper we study Hankel and Toeplitz operators on Dirichlet type spaces D. We obtain necessary and sufficient condition on the symbols for these operators to be bounded and to belong to the Schatten ideal S^p for certain and p.     

Finiteness of the discrete spectrum of some block Toeplitz operators

Sufficient conditions are given for the finiteness of the discrete spectrum of the block Toeplitz operatorT _A generated in the spaceH ₂ ⁿ by self-adjoint matrix functionA(t)(|t|=1). These results are obtained by means of theorems concerning the spectrum of a perturbed self-adjoint operators.     

Toeplitz-Hankel type operators on Dirichlet spaces

LetL ^,2 be the function space defined by the (weighted) Dirichlet energy integrals on the unit disk of complex plane. By constructing new orthogonal polynomials we give an orthogonal decomposition such thatA ₀ is just the (weighted) Dirichlet space. We define three kinds of Toeplitz and Hankel type operators, develop their boundedness andS _p -criteria, and reveal cut-off phenomena and Wu' phenomena.Research was supported in part by the National Natural Science Foundation of China.     

Toeplitz and Hankel type operators on the upper half-plane

An orthogonal decomposition of admissible wavelets is constructed via the Laguerre polynomials, it turns to give a complete decomposition of the space of square integrable functions on the upper half-plane with the measurey dxdy. The first subspace is just the weighted Bergman (or Dzhrbashyan) space. Three types of Ha-plitz operators are defined, they are the generalization of classical Toeplitz, small and big Hankel operators respectively. Their boundedness, compactness and Schatten-von Neumann properties are studied.Research was supported by the National Natural Science Foundation of China.     

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.     

Products of Toeplitz operators

An aspect of the theory of Toeplitz operators on generalised Hardy spaces is considered, namely, a necessary and sufficient condition on the symbols to ensure that the product of two Toeplitz operators is itself a Toeplitz operator. The answer to this question draws on many deep results of the theory of generalised Hardy spaces.     

Maximal inner spaces and Hankel operators on the Bergman space

The paper deals with two closely related questions about the Bergman space of the unit disk. First, we investigate a special class of invariant subspaces of the Bergman space, namely, invariant subspaces induced by certain Hankel operators. We show that such spaces always have the co-dimension 1 or 2 property; and we determine exactly when such a space has the co-dimension 1 property. Second, we introduce the notion of inner spaces in the Bergman space and give several characterizations of when an inner space is maximal.Research supported by the National Science Foundation     

Toeplitz operators with noncommuting symbols

LetM be a von Neumann algebra with a faithful normal tracial state and letH be a finite maximal subdiagonal subalgebra ofM. LetH ² be the closure ofH in the noncommutative Lebesgue spaceL ²(M). We consider Toeplitz operators onH ² whose symbol belong toM, and find that they possess several of the properties of Toeplitz operators onH ²( ) with symbol fromL ( ), including norm estimates, a Hartman-Wintner spectral inclusion theorem, and a characterisation of the weak^* continuous linear functionals on the space of Toeplitz operators.     

The finite section method for Moore-Penrose inversion of Toeplitz operators

It is shown that for certain classes of infinite block Toeplitz matricesT(a)=[a _j-k ] ₀ the Moore-Penrose inverses of the finite sectionT _n (a)=[a _j-k ] ₀ ^n–1 converge to the Moore-Penrose inverse ofT(a). Furthermore the convergence for modified finite section methods and the finite section method for Wiener-Hopf integral and related operators are studied.     

On operators commuting with Toeplitz operators modulo the finite rank operators

Let X be a bounded linear operator on the Hardy space H² of the unit disk. We show that if is of finite rank for every inner function θ, then X=T_?+F for some Toeplitz operator T_? and some finite rank operator F on H². This solves a variant of an open question where the compactness replaces the finite rank conditions.     

Subnormality and 2-hyponormality for Toeplitz operators

In this article we provide an example of a Toeplitz operator which is 2-hyponormal but not subnormal, and we consider 2-hyponormal Toeplitz operators with finite rank self-commutators.Supported by NSF research grant DMS-9800931.Supported by KOSEF research project No. R01-2000-00003.     

Toeplitz operators hyponormal with the unilateral shift

For the unilateral shift operator U on the Hardy space H²(T), we describe conditions on operators T, acting on H²(T), that are necessary and sufficient for the pair (U, T) to be jointly hyponormal. One necessary condition is that T be a Toeplitz operator. Consequently, we study certain nonanalytic symbols that give rise to Toeplitz operators hyponormal with the shift, and thereby obtain examples of noncommuting, jointly hyponormal pairs.Supported in part by a research grant from NSERC     

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.