Mini-Workshop: Operators on Spaces of Analytic Functions

Abstract

During the last twenty years there has been considerable activity and progress in the study of Banach spaces of analytic functions and their operators such as composition operators, multiplication operators and related natural classes of operators on these spaces. By use of the theory of analytic functions it is often possible to obtain very precise knowledge of the fine structure of these particular operators, and that has repercussions for our understanding of all bounded linear operators between Banach or Hilbert spaces. For example, the invariant subspace problem for separable Hilbert spaces can be reduced to the study of the invariant subspace structure of certain composition operators (E. Nordgren, P. Rosenthal and F. S. Wintrobe, 1987) or certain multiplication operators (C. Apostol, H. Bercovici, C. Foiac{s} and C. Pearcy, 1985).

Therefore, questions concerning invariant subspaces were one of the major topics in this mini-workshop. A. Atzmon considered in his first contribution scales of Banach spaces of analytic functions on which the lattice of invariant subspaces for the shift or the backward shift has no proper gaps. In his second talk he reported on recent progress concerning the translation invariant subspace problem on weighted ${\ell }^2$ spaces over $\mathbb{Z}$. A. Aleman and S. Richter investigated relations between the index $\dim M/zM$ of an invariant subspace $M$ for the operator of multiplication by the variable $z$ on Hilbert spaces of analytic functions on the unit disc $\mathbb{D}$ and the boundary behavior of the functions it contains, as well as the connection between abstract properties of this operator and the boundary behavior of functions in such Hilbert spaces of analytic functions.

Questions related to composition operators were the second major subject of the meeting. W. Smith gave an equivalent formulation of the Brennan conjecture concerning the derivative of conformal maps of the unit disc in terms of compactness of certain weighted composition operators. Th. W. Gamelin related the essential spectrum of composition operators on uniform algebras to the notion of hyperbolic boundedness. E. Gallardo–Gutiérrez focused on the problem of characterizing boundedness or compactness of composition operators on Hardy spaces $H^p(\Omega )$ over simply connected domains $\Omega$ in the complex plane. In particular she presented a complete characterization of symbols inducing bounded and compact composition operators on $H^p$-spaces over Lavrientev domains ($1\le p<\infty$). C. Sundberg considered natural topologies on the set of composition operators on Banach spaces of analytic functions on the unit disc, for which all composition operators induced by analytic maps $\phi :\mathbb{D}\to \mathbb{D}$ are bounded. M. Jury considered the $C^{\ast }$ algebra $C_{\Gamma }$ generated by the composition operators $C_{\phi }$, $\phi \in \Gamma$, $\Gamma$ a non-elementary Fuchsian group, and obtained index theorems for certain sums of weighted composition operators. His proofs involve the use of methods from noncommutative geometry.

Spectral properties of generalized Cesàro operators and weighted averaging operators on Hardy and weighted Bergman spaces were considered by E. Albrecht.

N. Feldman introduced the idea of interpolating measures for subnormal operators $S$, a natural generalization of the notion of bounded point evaluations. In the case that $S=M_z$ on $P^2(\mu )$ is pure, he gave a characterization of interpolating measures. He also included some illuminating examples. W. T. Ross showed that, for a finite, compactly supported positive measure $\mu$, the set of cyclic multiplication operators on $L^2(\mu )$ coincides with the set $S_{\mu }$ of all multiplication operators $M_{\varphi }$ on $L^2(\mu )$, $\varphi \in L_i^{\infty }(\mu )$, where $L_i^{\infty }(\mu )$ denotes the set of all $\mu$-essentially bounded functions that are one-to-one on a set of full $\mu$-measure. He also showed that $S_{\mu }$ has a common cyclic vector if and only if the continuous part of $\mu$ vanishes.

H. Jarchow discussed the problem of bounded extensions of bounded bilinear forms with an application to $H^{\infty }$ and the disc algebra.

R. Mortini gave in his talk a complete characterization of all countably generated prime ideals in $H^{\infty }$: A nonzero prime ideal $I$ in $H^{\infty }$ is countably generated if and only if either $I=\{f\in H^{\infty };f(z_0)=0\}$ for some $z_0\in \mathbb{D}$ or if $I$ is the ideal generated by the $n$-th roots of $S_{\sigma }(z)=\exp (-\frac{1+\bar{\sigma }z}{1-\bar{\sigma }z})$ for some $\sigma \in \partial \mathbb{D}$.

N. Nikolski showed that the condition numbers of the worst $N\times N$ matrices $A$ satisfying a Besov space $B_{p,q}^s$ functional calculus behave asymptotically as $N^s/\det (A)$ for $N\to \infty$. The proof depends on estimates of Besov analytic capacities of $N$ points subsets of the unit disc.

S. Shimorin discussed a new type of area theorems for univalent functions and, as a consequence, obtained a new series of sharp integral inequalities. His work was motivated by recent progress on Brennan's conjecture.

For a scale of spaces of analytic functions in $\mathbb{D}$ including the Korenblum space, A. Borichev obtained quantitative uniqueness theorems of the Lyubarski–Seip type.

Motivated by Birkhoff's ergodic theorem, F. Bayart and S. Grivaux have recently introduced an interesting new concept in hypercyclicity, that of frequently hypercyclic operators. In her talk, S. Grivaux gave some sufficient criteria in terms of Banach space properties of the underlying space. K.-G. Grosse–Erdmann showed that, for a frequently hypercyclic operator $T$, the operator $T\oplus T$ is hypercyclic. He also proved that an operator $T$ on the space of entire functions that commutes with the differentiation operator and that is not a multiple of the identity is frequently hypercyclic.

On Wednesday morning a problem session chaired by Th. W. Gamelin had been organized. Some of the problems discussed during that session are included at the end of this report. Further open questions were pointed out in many of the talks.

This mini-workshop was organized by Ernst Albrecht (Saarbrücken), Jean Esterle (Bordeaux), Raymond Mortini (Metz) and Stefan Richter (Knoxville). Unfortunately, Jean Esterle was unable to participate. All the participants were grateful for the hospitality and the stimulating atmosphere of the Forschungsinstitut Oberwolfach.