On the norm and the essential norm of weighted composition operators acting on the weighted Banach space of analytic functions

The objective of this paper is to use known properties of the associated weight to give new estimations of the norm and essential norm of the weighted com- position operators acting between weighted Banach spaces of analytic functions, in terms of composition with certain special functions in these spaces. As a consequence of our results, we obtain genuine extensions of a large number of recent results, and in particular, we extend classical results due to Montes-Rodríguez and Contreras and Hernández-Díaz.     

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.     

On an integral operator between Bloch-type spaces on the unit ball

We characterize the boundedness and compactness of the following integral-type operator

     

Beyond hyperinvariance for compact operators

We introduce a new class of operator algebras on Hilbert space. To each bounded linear operator a spectral algebra is associated. These algebras are quite substantial, each containing the commutant of the associated operator, frequently as a proper subalgebra. We establish several sufficient conditions for a spectral algebra to have a nontrivial invariant subspace. When the associated operator is compact this leads to a generalization of Lomonosov's theorem.     

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     

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that _limn→∞?[_an(_Cφ)]^1/n=e^{−1/Cap[φ(D)]}${\lim }_{n\to \infty }?{[a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}[\phi (\mathbb{D})]}$, where Cap[φ(D)]$\mathrm{Cap}[\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.     

Exceptional sets and Hilbert-Schmidt composition operators

It is shown that an analytic map ? of the unit disk into itself inducing a Hilbert-Schmidt composition operator on the Dirichlet space has the property that the set has zero logarithmic capacity. We also show that this is no longer true for compact composition operators on the Dirichlet space. Moreover, such a condition is not even satisfied by Hilbert-Schmidt composition operators on the Hardy space.     

Composition operators from large weighted Hardy spaces into the Dirichlet space

Given a large weighted Hardy space we show there exists a composition operatorC with that maps from that space into the unweighted Dirichlet space and lies in every Schatten p-class for 0<p<. This is in contrast to the situation in which the image space is a smaller weighted Dirichlet space. It is known that in that case it is not possible to find such a composition operator that is bounded.This research was supported in part by a summer stipend from Bellarmine College.     

Subnormality and composition operators on the Bergman space

Following the work of C. Cowen and T. Kriete on the Hardy space, we prove that under a regularity condition, all composition operators with a subnormal adjoint on A²(D) have linear fractional symbols of the form. Moreover, we show that all composition operators on the Bergman space having these symbols have a subnormal adjoint, with larger range for the parameterr than found in the Hardy space case.     

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.     

Essential norm of differences of weighted composition operators between weighted-type spaces on the unit ball

We find an asymptotically equivalent expression to the essential norm of differences of weighted composition operators between weighted-type spaces of holomorphic functions on the unit ball in C^N. As a consequence we characterize the compactness of these operators. The boundedness of these operators is also characterized.     

Generalized composition and Volterra type operators between QK spaces

Abstract

In this paper, the boundedness and compactness of the generalized composition operators and the products of Volterra type operators a nd composition operators between Q_K spaces are investigated. We also give a necessary condition for multiplication operators between Q_K spaces to be bounded or compact.     

Bounded and compact multipliers between Bergman and Hardy spaces

This paper studies the boundedness and compactness of the coefficient multiplier operators between various Bergman spacesA ^p and Hardy spacesH ^q . Some new characterizations of the multipliers between the spaces with exponents 1 or 2 are derived which, in particular, imply a Bergman space analogue of the Paley-Rudin Theorem on sparse sequences. Hardy and Bergman spaces are shown to be linked using mixed-norm spaces, and this linkage is used to improve a known result on (A ^p ,A ²), 1<p<2.Compact (H ¹,H ²) and (A ¹,A ²) multipliers are characterized. The essential norms and spectra of some multiplier operators are computed. It is shown that forp>1 there exist bounded non-compact multiplier operators fromA ^p toA ^q if and only ifpq.     

Algebraic and essentially algebraic composition operators onC(X)

Summary Given a compact Hausdorff spaceX, we may associate with every continuous mapa: X X a composition operatorC _a onC(X) by the rule(C _a f)(x) = f(a(x)). We describe all self-mapsa for whichC _a is an algebraic operator or an essentially algebraic operator (i.e. an operator algebraic modulo compact operators), determine the characteristic polynomialp _a (z) and the essentially characteristic polynomialq _a (z) in these cases and show how the connectivity ofX may be characterized in terms of the quotientsp _a (z)/q _a (z). Research supported by the Alfried Krupp Förderpreis für junge Hochschullehrer.     

Inverse spectral problems for Bessel operators

We study the inverse spectral problem for a class of Bessel operators given in L₂(0,1) by the differential expression

     

Norm-attaining compact operators

We show examples of compact linear operators between Banach spaces which cannot be approximated by norm attaining operators. This is the negative answer to an open question posed in the 1970s. Actually, any strictly convex Banach space failing the approximation property serves as the range space. On the other hand, there are examples in which the domain space has a Schauder basis.     

Essential norm of the difference of weighted composition operators

We study differences of weighted composition operators between weighted Banach spaces H _ν ^∞ of analytic functions with weighted sup-norms and give an expression for the essential norm of these differences. We apply our result to estimate the essential norm of differences of composition operators acting on Bloch-type spaces. Authors’ addresses: Mikael Lindstr?m, Department of Mathematics, Abo Akademi University, FIN 20500 Abo, Finland; Elke Wolf, Mathematical Institute, University of Paderborn, D-33095 Paderborn, Germany