On hyponormal Toeplitz operators with polynomial and circulant-type symbols

This paper characterises those hyponormal Toeplitz operators on the Hardy space of the unit circle among all Toeplitz operators that have polynomial symbols with circulant-type sets of coefficients.Supported in part by The Natural Sciences and Engineering Research Council of CanadaSupported in part by BSRI-96-1420 and KOSEF 94-0701-02-01-3     

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.     

On hyponormal toeplitz operators with polynomial and symmetric-type symbols

This paper treats the hyponormality of Toeplitz operators that have polynomial symbols with symmetric-type sets of coefficients.     

Hyponormality of Toeplitz operators with rational symbols

In this article we introduce a notion of `division' for rational functions and then give a criterion for hyponormality of (f, g are rational functions) in the cases where g divides f. Furthermore we show that we may assume, without loss of generality, that g divides f when we consider the hyponormality of . Supported in part by a grant from Faculty Research Fund, Sungkyunkwan University, 2004. Supported in part by a grant (R14-2003-006-01000-0) from the Korea Research Foundation.     

Toeplitz operators on the Fock space: Radial component effects

The paper is devoted to the study of specific properties of Toeplitz operators with (unbounded, in general) radial symbolsa=a(r). Boundedness and compactness conditions, as well as examples, are given. It turns out that there exist non-zero symbols which generate zero Toeplitz operators. We characterize such symbols, as well as the class of symbols for whichT _a =0 impliesa(r)=0 a.e. For each compact setM there exists a Toeplitz operatorT _a such that spT _a =ess-spT _a =M. We show that the set of symbols which generate bounded Toeplitz operators no longer forms an algebra under pointwise multiplication.Besides the algebra of Toeplitz operators we consider the algebra of Weyl pseudodifferential operators obtained from Toeplitz ones by means of the Bargmann transform. Rewriting our Toeplitz and Weyl pseudodifferential operators in terms of the Wick symbols we come to their spectral decompositions.This work was partially supported by CONACYT Project 27934-E, México.The first author acknowledges the RFFI Grant 98-01-01023, Russia.     

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.     

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.     

On Toeplitz operators with quasihomogeneous symbols

In this paper, we give some basic results concerning Toeplitz operators whose symbol is of the form e^{i p θ}ϕ, where ϕ is a radial function, then use these results to characterize all Toeplitz operators which commute with them.Received: 12 June 2004; revised: 27 January 2005     

On unbounded hyponormal operators II

The paper deals with unbounded hyponormal operators. Among others it is proved that any closed hyponormal operator with spectrum contained in a parabola generates a cosine function.     

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.     

Semi-commutators of Toeplitz operators on the Bergman space

In this paper several necessary and sufficient conditions are obtained for the semi-commutator of Toeplitz operators andT _g with bounded pluriharmonic symbols on the unit ball to be compact on the Bergman space. Using -harmonic function theory on the unit ball we show that with bounded pluriharmonic symbolsf andg is zero on the Bergman space of the unit ball or the Hardy space of the unit sphere if and only if eitherf org is holomorphic.The author was supported in part 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.     

Toeplitz operators defined by sesquilinear forms: Fock space case

The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions.     

Algebras on the unit disk and Toeplitz operators on the Bergman space

This paper studies algebras of functions on the unit disk generated byH ^∞(D) and bounded harmonic functions. Using these algebras, we characterize compact semicommutators and commutators of Toeplitz operators with harmonic symbols on the Bergman space. Supported in part by the National Science Foundation and the University Research Council of Vanderbilt University.     

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.     

Toeplitz operators on Dirichlet spaces

In this paper we consider Toeplitz operators on Dirichlet spaces of the unit disk in whose symbols are nonnegative measures. We obtain necessry and sufficient conditions on the symbols for the operator to be bounded and compact. If the symbols are supported in a cone we also get necessary and sufficient conditions for the operators to belong to the Schatten p-class. Application to the Hankel operators are indicated.This work supported in part by NSF grant DMS 8701271     

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.