Weakly supercyclic operators

Inspired by a recent result that a weakly hypercyclic operator may fail to be norm hypercyclic, we show there exists a weakly supercyclic operator that fails to be norm supercyclic. Moreover, despite a classical result of Hilden and Wallen that every unilateral weighted backward shift is supercyclic, we show such an operator may have a weakly supercyclic vector that is not a norm supercyclic vector. In addition to these results, we extend a result of Kitai by showing a hyponormal operator cannot be weakly hypercyclic.     

Backward Aluthge Iterates of a Hyponormal Operator Have Scalar Extensions

The backward Aluthge iterate (defined below) of a hyponormal operator was initiated in [11]. In this paper we characterize the backward Aluthge iterate of a weighted shift. Also we show that the backward Aluthge iterate of a hyponormal operator has an analogue of the single valued extension property for . Finally, we show that backward Aluthge iterates of a hyponormal operator have scalar extensions. As a corollary, we get that the backward Aluthge iterate of a hyponormal operator has a nontrivial invariant subspace if its spectrum has interior in the plane.     

Generalized Appell bases

In the context of Köthe spaces we study the bases related with the backward unilateral weighted shift operator, the so-called generalized derivation operator, extending known results for spaces of analytic functions. These bases are a subclass of Sheffer sequences called generalized Appell sequences and they are closely connected with the isomorphisms invariant by the weighted shift. We use methods of the non classical umbral calculi to give conditions for a generalized Appell sequence to be a basis.     

Dilation to the unilateral shifts

The classical result of Foias says that an operator power dilates to a unilateral shift if and only if it is aC ₀ contraction. In this paper, we consider the corresponding question of dilating to a unilateral shift. We show tht for contractions with at least one defect index finite, dilation and power dilation to some unilateral shift amount to the same thing. The only difference is on the minimum multiplicity of the unilateral shift to which the contraction can be (power) dilated. We also obtain a characterization of contractions which are finite-rank perturbations of a unilateral shift, generalizing the rank-one perturbation result of Nakamura.     

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     

Compressions of Stable Contractions

The stability of compressions of stable contractions is studied and a sufficient orbit condition is given. On the other hand, it is shown that there are non-stable compressions of the 1-dimensional backward shift and a complete characterization of weighted unilateral shifts with this property is provided. Dilations of bilateral weighted shifts to backward shifts are also considered.     

Common supercyclic vectors for a path of operators

We examine common supercyclic vectors for a path of operators. In particular, we show that the path consisting of convex combinations of two arbitrary unilateral weighted backward shifts has a dense Gδ set of common supercyclic vectors. Moreover, we show there exists a path with a dense Gδ set of common supercyclic vectors between a unilateral weighted backward shift which satisfies the Supercyclicity Criterion, and an operator which does not. Lastly, we provide an example of a path of unilateral weighted backward shifts that fails to have a common supercyclic vector.     

w-Hyponormal Operators have Scalar Extensions

In this paper we show that every w-hyponormal operator has a scalar extension, i.e. is similar to the restriction to an invariant subspace of a scalar operator of order 4. As a corollary, we obtain that every w-hyponormal operator satisfies the property (β).     

Extensions of normal operators

We prove that every one dimensional extension of a separably acting normal operator has a cyclic commutant, and that every non-algebraic normal operator has a two-dimensional extension which fails to have a cyclic commutant. Contrasting this, we prove that ifT is an extension of a normal operator by an algebraic operator then the weakly closed algebraW(T) has a separating vector.Partially supported by NSF Grant DMS-9107137     

On Rosenthal-Shields' problem

In this paper it will be shown that the spectrum of every unicellular unilateral weighted shift operator on a symmetric Banach space is the singleton set {0}. From this, we give an affirmative answer to Rosenthal-Shields' problem. Project supported by the National Natural Science Foundation of China     

The homogeneous shifts

A bounded linear operator T on a complex Hilbert space is called homogeneous if the spectrum of T is contained in the closed unit disc and all bi-holomorphic automorphisms of this disc lift to automorphisms of the operator modulo unitary equivalence. We prove that all the irreducible homogeneous operators are block shifts. Therefore, as a first step in classifying all of them, it is natural to begin with the homogeneous scalar shifts.In this paper we determine all the homogeneous (scalar) weighted shifts. They consist of the unweighted bilateral shift, two one-parameter families of unilateral shifts (adjoints of each other), a one-parameter family of bilateral shifts and a two-parameter family of bilateral shifts. This classification is obtained by a careful analysis of the possibilities for the projective representation of the Möbius group associated with an irreducible homogeneous shift.     

Trace class backward weighted shifts are quasisubscalar

Recently, the author generalized Putinar techniques. In this paper we use those recent techniques and results to show (Theorem 3.1) that every trace class backward weighted shift with a monotone decreasing weight sequence is quasisubscalar.

     

Reducing subspaces of compressed analytic Toeplitz operators on the Hardy space

In this paper we discuss necessary conditions and sufficient conditions for the compression of an analytic Toeplitz operator onto a shift coinvariant subspace to have nontrivial reducing subspaces. We give necessary and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial and obtain examples of reducing subspaces from these kernels. Motivated by this result we give necessary conditions and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial in terms of the supports of the two inner functions. By studying the commutant of a compression, we are able to give a necessary condition for the existence of reducing subspaces on certain shift coinvariant subspaces.     

REAL AND COMPLEX OPERATOR IDEALS

Abstract

The powerful concept of an operator ideal on the class of all Banach spaces makes sense in the real and in the complex case. In both settings we may, for example, consider compact, nuclear, or 2-summing operators, where the definitions are adapted to each other in a natural way. This paper deals with the question whether or not that fact is based on a general philosophy. Does there exists a one-to-one correspondence between “real properties” and “complex properties” defining an operator ideal? In other words, does there exist for every real operator ideal a uniquely determined corresponding complex ideal and vice versa?

Unfortunately, we are not abel to give a final answer. Nevertheless, some preliminary results are obtained. In particular, we construct for every real operator ideal a corresponding complex operator ideal and for every complex operator ideal a corresponding real one. However, we conjecture that there exists a complex operator ideal which can not be obtained from a real one by this construction.

The following approach is based on the observation that every complex Banach space can be viewed as a real Banach space with an isometry acting on it like the scalar multiplication by the imaginary unit i.     

Genericity of wild holomorphic functions and common hypercyclic vectors

Let T_α be the translation operator by α in the space of entire functions defined by . We prove that there is a residual set G of entire functions such that for every f∈G and every the sequence is dense in , that is, G is a residual set of common hypercyclic vectors ( functions) for the family . Also, we prove similar results for many families of operators as: multiples of differential operator, multiples of backward shift, weighted backward shifts.     

q-deformed Circular Operators

In this paper, q-deformed circularity of an operator in a Hilbert space is introduced and investigated. In particular, the q-deformed circularity of a weighted shift is characterized by an integer, which is uniquely determined by the operator. It is shown that that the spectrum of a q-deformed circular weighted shift is the whole complex plane.     

A note on integral representations of linear operators

In this paper we describe families of those bounded linear operators that are simultaneously unitarily equivalent to integral operators with smooth Carleman kernels. The singleton case of the main result implies that every integral operator is unitarily equivalent to an integral operator having smooth Carleman kernel.     

Operator theory in the Hardy space over the bidisk (II)

This paper is a continuation of a project of developing a systematic operator theory inH ²(D ²). A large part of it is devoted to a study ofevaluation operator which is a very useful tool in the theory. A number of elementary properties of the evaluation operator are exhibited, and these properties are used to derive results in other topics such as interpretation of characteristic opertor function inH ²(D ²), spectral equivalence, compactness, compressions of shift operators, etc., Even though some results reflect the two variable nature ofH ²(D ²), the goal of this paper is to manifest a close tie between the operator theory inH ²(D ²) and classical single operator theory. The unilateral shift of a finite multiplicity and the Bergman shift will be used as examples to illustrate some of the results.Research in this paper is partially supported by a grant from the national science foundation DMS 9970932.     

算子权移位的Banach约化性

本文给出了单边算子权移位是ＢＲ算子的充要条件及双边算子权移位是ＢＲ算子的充分条件．做为推论，重新得到了［１］中的一个结果，并得到了一类ＢＩＲ单边算子权移位的例子。最后，给出大量的ＨＩＲ但同时又是ＢＲ的单边算子权移位。     

Dual analytic models of seminormal operators

In [6] (after Clancey's work [2]), Martin and Putinar introduced their two-dimensional functional model of a hyponormal operator, which reduces it to the multiplication by the independent variable in a space of distributions. Here we define another model which does (almost) the same for the adjoint operator. We also explain a close relation between these two models and dual bundle shift models of linear operators introduced in [13]. As application, an estimate of the effectual rational multiplicity of hyponormal operators is given.The research described in this publication was made possible in part by Grant No. NW8000 from the International Science Foundation