Mikael Lindström

COMPOSITION OPERATORS BETWEEN WEIGHTED BANACH SPACES OF ANALYTIC FUNCTIONS

We characterize those analytic self-maps ’ of the unit disc which generate bounded or compact composition operators C’ between given weighted Banach spaces H 1 v or H 0 v of analytic functions with the weighted sup-norms. We characterize also those composition operators which are bounded or compact with respect to all reasonable weights v.

DIFFERENCES OF COMPOSITION OPERATORS BETWEEN WEIGHTED BANACH SPACES OF HOLOMORPHIC FUNCTIONS

We consider differences of composition operators between given weighted Banach spaces H∞ v or H 0 v of analytic functions with weighted sup-norms and give estimates for the distance of these differences to the space of compact operators. We also study boundedness and compactness of the operators. Some examples illustrate our results. MSC 2000: 47B33, 47B38.

Spectra of weighted composition operators on weighted banach spaces of analytic functions

We determine the spectra of weighted composition operators acting on the weighted Banach spaces of analytic functionsHνp∞ when the symbolφ has a fixed point in the open unit disk. Further, we apply this result to give the spectra of composition operators on Bloch type spaces. In particular, we answer in the affirmative a conjecture by MacCluer and Saxe.

Isometric weighted composition operators on weighted Banach spaces of type

We characterize those weighted composition operators on weighted Banach spaces of holomorphic functions of type H ∞ which are an isometry.

Connected components in the space of composition operators onH∞ functions of many variables

LetE be a complex Banach space with open unit ballBe. The structure of the space of composition operators on the Banach algebra H∞, of bounded analytic functions onBe with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly insideBe form a path connected component. WhenE is a Hilbert space or aCo(X)- space, the path connected components are shown to be the open balls of radius 2.

Uniform factorization for compact sets of operators

We prove a factorization result for relatively compact subsets of compact operators using the Bartle and Graves Selection Theorem, a characterization of relatively compact subsets of tensor products due to Grothendieck, and results of Figiel and Johnson on factorization of compact operators. A further proof, essentially based on the Banach-Dieudonné Theorem, is included. Our methods enable us to give an easier proof of a result of W.H. Graves and W.M. Ruess.

The Essential Norm of a Bloch-to-Qp Composition Operator

The Qp spaces coincide with the Bloch space for p > 1 and are subspaces of BMOA for 0 < p ≤ 1. We obtain lower and upper estimates for the essential norm of a composition operator from the Bloch space into Qp, in particular from the Bloch space into BMOA.

A general approach to infinite-dimensional holomorphy

In this paper we present a general theory for holomorphic functions which is based on continuous convergence instead of topologies. The theory can be applied to locally convex spaces and bornological spaces.

Essential Norm of Operators into Weighted-Type Spaces on the Unit Ball

The essential norm of any operator from a general Banach space of holomorphic functions on the unit ball in ℂ𝑛 into the little weighted-type space is calculated. Some applications of the formula are given.

Spectra of composition operators on algebras of analytic functions on Banach spaces

Let E be a Banach space, with unit ball B E . We study the spectrum and the essential spectrum of a composition operator on H ∞ ( B E ) determined by an analytic symbol with a fixed point in B E . We relate the spectrum of the composition operator to that of the derivative of the symbol at the fixed point. We extend a theorem of Zheng to the context of analytic symbols on the open unit ball of a Hilbert space.