Hyperinvariant subspaces ofC 0-operators over a multiply connected region

In this paper we find a relation between the lattice of hyperinvariant subspaces of an operatorT of classC ₀ over a multiply connected region and that of its Jordan modelT. It is shown that, generally, the lattice corresponding toT can be identified with a retract of that corresponding toT. Thus the Jordan model has the smallest lattice of hyperinvariant subspaces in a given quasisimilarity class.     

On invariant subspaces of Volterra-type operators

The lattice of all the closed, invariant subspaces of the Volterra integration operator onL ²[0, 1] is equal to {B(a):a[0, 1]}, whereB(a)={fL ²[0, 1]:f=0 a.e. on [0,a]}. In order to extend this result to Banach function spaces we study the Volterra-type operatorV that was introduced in [7] for the case ofL ^p -spaces. Our main result characterizesL-closed subspaces of a Banach function spaceL that are invariant underV, whereL denotes the associate space ofL. In particular, if the norm ofL is order continuous and ifV is injective, then all the closed, invariant subspaces ofV are determined.This work was supported by the Research Ministry of Slovenia.     

L 1 factorizations and invariant subspaces for weighted composition operators

For a contraction operator T with spectral radius less than one on a Banach space , it is shown that the factorization of certain L¹ functions by vectors x in and x*. in , in the sense that for n ≧ 0, implies the existence of invariant subspaces for T. Explicit formulae for such factorizations are given in the case of weighted composition operators on reproducing kernel Hilbert spaces. An interpolation result of McPhail is applied to show how this can be used to construct invariant subspaces of hyperbolic weighted composition operators on H². Received: 1 November 2005     

Reproducing kernels for Hardy spaces on multiply connected domains

Associated with a boundedg-holed (g0) planar domainD are two types of reproducing kernel Hilbert spaces of meromorphic functions onD. We give explicit formulas for the reproducing kernel functions of these spaces. The formulas are in terms of theta functions defined on the Jacobian variety of the Schottky double of the regionD. As applications we settle a conjecture of Abrahamse concerning Nevalinna-Pick interpolation on an annulus and obtain explicit formulas for the curvature (in the sense of Cowen and Douglas) of rank 1 bundle shift operators.     

On the connected component of compact composition operators on the Hardy space

We show that there exist non-compact composition operators in the connected component of the compact ones on the classical Hardy space H². This answers a question posed by Shapiro and Sundberg in 1990. We also establish an improved version of a theorem of MacCluer, giving a lower bound for the essential norm of a difference of composition operators in terms of the angular derivatives of their symbols. As a main tool we use Aleksandrov-Clark measures.     

Weak compactness in certain star-shift invariant subspaces

The context of much of the work in this paper is that of a backward-shift invariant subspace of the form , where B is some infinite Blaschke product. We address (but do not fully answer) the question: For which B can one find a (convergent) sequence in K_B such that the sequence of real measures converges weak-star to some nontrivial singular measure on ? We show that, in order for this to hold, K_B must contain functions with nontrivial singular inner factors. And in a rather special setting, we show that this is also sufficient. Much of the paper is devoted to finding conditions (on B) that guarantee that K_B has no functions with nontrivial singular inner factors. Our primary result in this direction is based on the “geometry” of the zero set of B.     

On certain quotients of Hardy spaces

The quotient space of a Hardy space on a half-plane Imz> modulo the subspace of elements containing a factore ^iz is in some sense independent of . A formula is derived which exhibits a correspondence between any two such quotient spaces.Supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine.     

Common invariant subspaces for collections of operators

Let be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that is finitely quasinilpotent at a vectorx ₀X whenever for any finite subset of the joint spectral radius of atx ₀ is equal 0. If such collection contains a non-zero compact operator, then and its commutant have a common non-trivial invariant, subspace. If in addition, is a collection of positive operators on a Banach lattice, then has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then has a common non-trivial invariant closed ideal.This work was supported by the Research Ministry of Slovenia.     

A cross section from invariant subspaces to inner functions

Beurling's well known theorem connects the study of invariant subspaces to that of inner functions over the unit disc. In this paper, we will further explore this connection and, as a corollary of the result, show a one to one correspondence between the components of the invariant subspace lattice and the components of the space of inner functions.     

Shift invariant spaces inL 2

If is a complex, separable Hilbert space, letL ² () denote theL ²-space of functions defined on the unit circle and having values in . The bilateral shift onL ²() is the operator (U f)()=f(). A Hilbert spaceH iscontractively contained in the Hilbert spaceK ifHK and the inclusion mapH–K is a contraction. We describe the structure of those Hilbert spaces, contractively contained inL ²(), that are carried into themselves contractively byU . We also do this for the subcase of those spaces which are carried into themselves unitarily byU .     

Weighted Composition Operators on H 2 and Applications

Operators on function spaces acting by composition to the right with a fixed selfmap φ of some set are called composition operators of symbol φ. A weighted composition operator is an operator equal to a composition operator followed by a multiplication operator. We summarize the basic properties of bounded and compact weighted composition operators on the Hilbert Hardy space on the open unit disk and use them to study composition operators on Hardy–Smirnov spaces. Submitted: January 30, 2007. Revised: June 19, 2007. Accepted: July 11, 2007.     

An invariant subspace problem forp=1 Bergman spaces on slit domains

In this paper, we characterize thez-invariant subspaces that lie between the Bergman spacesA ¹ (G) andA ¹ (G/K), whereG is a bounded region in the complex plane andK is a compact subset of a simple arc of classC ¹.     

On a Theorem of Godefroy and Shapiro

G. Godefroy and J. H. Shapiro have shown that every operator on , that commutes with all translation operators , and that is not a scalar multiple of the identity is hypercyclic. We show that they are even frequently hypercyclic. In addition, we obtain growth conditions that may be satisfied by corresponding frequently hypercyclic entire functions.     

Topological Structure of the Space of Weighted Composition Operators on H ∞

We study properties of the topological space of weighted composition operators on the space of bounded analytic functions on the open unit disk in the uniform operator topology. Moreover, we characterize the compactness of differences of two weighted composition operators.     

Normal boundary dilations and rationally invariant subspaces

The first named author was supported by grants from the National Science Foundation.     

Holomorphic Operators Generating Dense Images

The existence of infinite dimensional closed linear spaces of holomorphic functions f on a domain G in the complex plane such that Tf has dense images on certain subsets of G, where T is a continuous linear operator, is analyzed. Necessary and sufficient conditions for T to have the latter property are provided and applied to obtain a number of concrete examples: infinite order differential operators, composition operators and multiplication operators, among others. This work was supported in part by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 and by MEC DGES Grants MTM2006-13997-C02-01 and MTM2004-21420-E.     

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.     

SOME FUNCTIONAL ANALYTIC PROPERTIES OF COMPOSITION OPERATORS

Abstract

This is a report on a number of recent results on composition operators which map, for 0 < p ? q ∞, the Hardy space H^p (on the unit disk in the complex plane) into H ^q. Attention is focused on questions of boundedness (existence), compactness, order boundedness and, in connection with the latter, on relating the absolutely summing and nuclearity character as well as special factorization properties of the operator to function theoretic properties of the defining symbol. Moreover, tools are provided to show that certain classes of operators can well be distinguished already on the level of composition operators.