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     

Block toeplitz operators with rational symbol analytic at infinity

This paper concerns the problem of explicit inversion of a block Toeplitz operator with rational and analytic at infinity symbol. The necessary and sufficient conditions for the invertibility and explicit formulas for the inverse are given in terms of the realization of the symbol.     

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).     

Commuting Toeplitz operators on the Hardy space of the bidisk

On the Hardy space of the bidisk, we give a necessary and sufficient condition for a bounded symbol of a Toeplitz operator that commutes with another Toeplitz operator whose symbol is a certain type of bounded symbol.     

多圆盘Bergman空间上Toeplitz算子的乘积和交换性

该文给出了多圆盘Bergman空间上两个带有某种符号的Toeplitz算子的乘积等于另一个Toeplitz算子的充分必要条件,并且给出了乘积算子所带符号的公式.接下来,相应的研究了它的交换性.这些研究结果都是根据符号函数的Mellin变换.     

Lax-Phillips Scattering for Atomorphic Functions Based on the Eisenstein Transform

We construct a Lax-Phillips scattering system on the arithmetic quotient space of the Poincaré upper half-plane by the full modular group, based on the Eisenstein transform. We identify incoming and outgoing subspaces in the ambient space of all functions with finite energy-form for the non-Euclidean wave equation. The use of the Eisenstein transform along with some properties of the Eisenstein series of two variables enables one to work only on the space corresponding to the continuous spectrum of the Laplace-Beltrami operator. It is shown that the scattering matrix is the complex function appearing in the the functional equation of the Eisenstein series of two variables. We obtain a compression operator constructed from the Laplace-Beltrami operator, whose spectrum consists of eigenvalues that coincide, counted with multiplicities, with the non-trivial zeros of the Riemann zeta-function. For this purpose we construct and use a scattering model on the one-dimensional Euclidean space.      

Decompositions of a Krein space in regular subspaces invariant under a uniformly bounded C0-semigroup of bi-contractions

We give necessary and sufficient conditions under which a C₀-semigroup of bi-contractions on a Krein space is similar to a semigroup of contractions on a Hilbert space. Under these and additional conditions we obtain direct sum decompositions of the Krein space into invariant regular subspaces and we describe the behavior of the semigroup on each of these summands. In the last section we give sufficient conditions for the co-generator of the semigroup to be power bounded.     

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.     

Reducing Subspaces of Toeplitz Operators on Nψ-type Quotient Modules on the Torus

In this paper, we prove that the Toeplitz operator with finite Blaschke product symbol Sψ(z) on Nψhas at least m non-trivial minimal reducing subspaces, of Sψ(z) on any of these minimal reducing subspaces is unitary equivalent to the Bergman shift Mz.     

The Fredholm Index for Elements of Toeplitz-Composition C<Superscript>*</Superscript>-Algebras

We analyze the essential sectrum and index theory of elements of Toeplitz-composition C^*-algebras (algebras generated by the Toeplitz algebra and a single linear-fractional composition operator, acting on the Hardy space of the unit disk). For automorphic composition operators we show that the quotient of the Toeplitz-composition algebra by the compacts is isomorphic to the crossed product C^*-algebra for the action of the symbol on the boundary circle. Using this result we obtain sufficient conditions for polynomial elements of the algebra to be Fredholm, by analyzing the spectrum of elements of the crossed product. We also obtain an integral formula for the Fredholm index in terms of a generalized Chern character. Finally we prove an index formula for the case of the non-parabolic, non-automorphic linear fractional maps studied by Kriete, MacCluer and Moorhouse.     

Weyl's theorem and quasi-similarity

In this paper we give necessary and sufficient conditions of a finitely ascensive operator to obey Weyl's theorem and study Weyl's theorem for quasi-affine transforms. In particular, we give an improvement of the earlier result of Duggal forp-hyponormal operators.     

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     

Symbols of truncated Toeplitz operators

We consider three topics connected with coinvariant subspaces of the backward shift operator in Hardy spaces Hp:

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properties of truncated Toeplitz operators;

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Carleson-type embedding theorems for the coinvariant subspaces;

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factorizations of pseudocontinuable functions from H¹.

These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by D. Sarason.     

Toeplitz operators with frequency modulated semi-almost periodic symbols

It is well known that amplitude modulation does not affect Fredholmness of Toeplitz operators. The same is true for frequency modulation provided the symbol of the operator is piecewise continuous. In this article, it is shown that frequency modulation can destroy Fredholmness for Toeplitz operators with almost periodic symbols; the corresponding example is based on the observation that certain almost periodic functions become semi-almost periodic functions after appropriate frequency modulation. Moreover, this article contains several results that can be employed in order to decide whether a Toeplitz operator with a frequency modulated semi-almost periodic symbol is Fredholm.     

Bergman空间上拟齐次Toeplitz算子的乘积

使用Mellin变换作为工具,讨论了Bergman空间上以拟齐次函数为符号的Toeplitz算子的乘积问题,得出了当拟齐次函数的度处于三种不同情况时两个Toeplitz算子乘积仍是Toeplitz算子的充分必要条件.     

Cellular-indecomposable subnormal operators III

A bounded operatorT is called cellular-indecomposable ifL M {0} wheneverL andM are nonzero invariant subspaces forT. We prove that a cyclic subnormal operator is cellular-indecomposable if and only if it is quasi-similar to an analytic Toeplitz operator whose symbol is a weak-star generator ofH . This completes our previous work [5], [6].     

Reducing Subspaces of Toeplitz Operators on $N_ϕ$-Type Quotient Modules on the Torus

In this paper, we prove that the Toeplitz operator with finite Blaschke product symbol $S_{ψ(z)}$ on $N_ϕ$ has at least $m$ non-trivial minimal reducing subspaces, where $m$ is the dimension of $H^2(Γ_ω) ⊖ ϕ(ω)H^2 (Γ_ω)$. Moreover, the restriction of $S_{ψ(z)}$ on any of these minimal reducing subspaces is unitary equivalent to the Bergman shift $M_z$.     

块对偶Toeplitz算子的乘积

本文刻画了向量值Bergman空间上块对偶Toeplitz算子有界性和紧性,给出了块对偶Toeplitz算子的乘积是块对偶Toeplitz算子的充要条件.     

多重调和Bergman空间上径向Toeplitz算子的交换性

In this paper, we study the commutativity of Toeplitz operators with radial symbols on the pluriharmonic Bergman space. We obtain the necessary and sufficient conditions for the commutativity of bounded Toeplitz operator and Toeplitz operator with radial symbol on the pluriharmonic Bergman space.     

Kernels of Hankel operators and hyponormality of Toeplitz operators

We give a formula for and describe when . We explore the hyponormality of Toeplitz operators whose symbols are of circulant type and some more general types. In addition, we discuss formulas for and estimates of the rank of the self-commutator of a hyponormal Toeplitz operator. Received September 17, 1999 / Revised May 25, 2000 / Published online December 8, 2000