The spectra of some algebras of analytic mappings (original) (raw)

On the spectra of algebras of analytic functions

Contemporary Mathematics, 2012

In this paper we survey the most relevant recent developments on the research of the spectra of algebras of analytic functions. We concentrate mainly in three algebras, the Banach algebra H ∞ (B) of all bounded holomorphic functions on the unit ball B of a complex Banach space X, the Banach algebra of the ball Au(B), and the Fréchet algebra H b (X) of all entire functions bounded on bounded sets.

Spectra of Algebras of Entire Functions on Banach Spaces

Proceedings of the American Mathematical Society

We obtain an explicit description of the spectrum (set of closed maximal ideals) of Hb(X)H_{b}(X)Hb(X) , algebra of analytic functions on a Banach space X which are bounded on bounded subsets. We show that the spectrum of Hb(X)H_{b}(X)Hb(X) admits a natural linear structure. Some applications to the algebra of uniformly continuous and bounded analytic functions on the unit ball BsubsetXB \subset XBsubsetX are indicated.

The spectra of algebras of Lorch analytic mappings

Topology, 2009

For a complex Banach algebra E, let H L (E) be the space of the mappings from E into E that are analytic in the sense of Lorch. We will show that H L (E) is a closed subalgebra of H b (E, E) and we will give a description of its spectrum. As an application we will show that E is semi-simple if and only if H L (E) is semi-simple. We will also give descriptions of the spectra of other algebras of analytic mappings in the sense of Lorch. In particular we will study the spectrum of the Banach algebra H ∞ L (int B E) of the bounded mappings from int B E into E that are analytic in the sense of Lorch.

The approximation property for spaces of holomorphic functions on infinite-dimensional spaces I

Journal of Approximation Theory, 2004

For an open subset U of a locally convex space E; let ðHðUÞ; t 0 Þ denote the vector space of all holomorphic functions on U; with the compact-open topology. If E is a separable Fre´chet space with the bounded approximation property, or if E is a (DFC)-space with the approximation property, we show that ðHðUÞ; t 0 Þ has the approximation property for every open subset U of E: These theorems extend classical results of Aron and Schottenloher. As applications of these approximation theorems we characterize the spectra of certain topological algebras of holomorphic mappings with values in a Banach algebra. r

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

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2009

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 ∞ (BE) determined by an analytic symbol with a fixed point in BE. 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.

Approximately spectrum-preserving maps

Journal of Functional Analysis, 2011

Let X and Y be superreflexive complex Banach spaces and let B(X) and B(Y) be the Banach algebras of all bounded linear operators on X and Y , respectively. If a bijective linear map Φ : B(X) → B(Y) almost preserves the spectra, then it is almost multiplicative or anti-multiplicative. Furthermore, in the case where X = Y is a separable complex Hilbert space, such a map is a small perturbation of an automorphism or an anti-automorphism.

Compact holomorphic mappings on Banach spaces and the approximation property

Journal of Functional Analysis, 1976

1. Let E be a complex Banach space. It is well known that C(E\C), the space of continuous scalar-valued functions on E endowed with the compact-open topology, always has the approximation property, since there are continuous partitions of unity. However, for the space H{E\C) of holomorphic scalar-valued functions on E 9 the situation is more complicated. In §2 of this note, we describe this situation. Briefly, there is an exact analogy between the question of approximation by finite rank linear mappings on compact sets and the question of approximation by finite rank holomorphic mappings on compact sets.

Spectral properties of elements in different Banach algebras

Glasgow Mathematical Journal, 1991

A be a Banach algebra with unit 1 and let B be a Banach algebra which is a subalgebra of A and which contains 1. In [5] Barnes gave sufficient conditions for B to be inverse closed in A. In this paper we consider single elements and study the question of how the spectrum relative to B of an element in B relates to the spectrum of the element relative to A.