Heinrich Raubenheimer

More Spectral Theory in Ordered Banach Algebras

We continue our development of spectral theory for positive elements in an ordered Banach algebra. In particular we provide a suitable version of the Krein-Rutman theorem, obtain some results concerning the peripheral spectrum of a positive element and provide a characterisation of positive quasi inessential elements, in the context of an ordered Banach algebra.

Spectral properties of elements in different Banach algebras

by J. J. GROBLER and H. RAUBENHEIMER(Received 7 June, 1989)1. Introduction. Let A be a Banach algebra with unit 1 and let B be a Banachalgebra which is a subalgebra of A and which contains 1. In [5] Barnes gave sufficientconditions for B to be inverse closed in A. In this paper we consider single elements andstudy the question of how the spectrum relative to B of an element in B relates to thespectrum of the element relative to A.This question has been studied by a number of authors (see [14, 2, 3, 4, 12, 13, 15])under various conditions, usually requiring the norm on B to be finer than the norm on Aand in some cases requiring B to be a Banach algebra of operators on a Banach lattice.We shall merely assume the algebra B to be a subalgebra of A and we shall not assumeany relationship between the norms of A and B. Of course, if B is semi-simple and closedin A, the embedding of B into A is continuous (see [7, Theorem 25.9]). We do not intendto enter into a discussion of the question whether or not this is always true—this is a deepquestion which seems also to be related to the model of set theory used (see [8]). Thepoint we want to emphasize is that the main results are independent of the continuity ofthe embedding.If A is an algebra we denote the spectrum of an element a eA by o(a, A) and itsspectral radius by

Ruston elements and fredholm theory relative to arbitrary homomorphisms

We extend some of the Fredholm theory by, among other things, developing the theory of Ruston and almost Ruston elements and spectra relative to an arbitrary homomorphism. In addition, we provide a number of applications and generalise certain well-known results.

On regularities and Fredholm theory

We investigate the relationship between the regularities and the Fredholm theory in a Banach algebra.

THE o-SPECTRUM OF r-ASYMPTOTICALLY QUASI-FINITE-RANK OPERATORS

Abstract We introduce the class of r-asymptotically quasi finite rank operators (which properly contains the class of r-compact operators) and show that for these operators the spectrum and o-spectrum coincide.

The index for Fredholm elements in a Banach algebra via a trace

We show that the index defined via a trace for Fredholm elements in a Banach algebra has the property that an index zero Fredholm element can be decomposed as the sum of an invertible element and an element in the socle. We identify the set of index zero Fredholm elements as an upper semiregularity with the Jacobson property. The Weyl spectrum is then characterized in terms of the index.

r-ASYMPTOTICALLY QUASI-FINITE RANK OPERATORS AND THE SPECTRUM OF MEASURES

Abstract The r-asymptotically quasi finite rank operators were introduced in [10]. For regular operators on Banach lattices, these operators are the order theoretic analogue of Riesz operators on Banach spaces. We establish their basic properties and apply these in the spectral analysis of convolution operators.

On regular Riesz operators

The r-asymptotically quasi finite rank operators on Banach lattices are examples of regular Riesz operators. We characterise Riesz elements in a subalgebra of a Banach algebra in terms of Riesz elements in the Banach algebra. This enables us to characterize r-asymptotically quasi finite rank operators in terms of adjoint operators. The r-asymptotically quasi finite rank operators are also employed to study the following problem: Suppose operators S and T on a Banach lattice E satisfy 0 ≤ S ≤ T. If T is a Riesz operator, when is it true that S is a Riesz operator?

FINITE RANK RIESZ OPERATORS

We provide conditions under which a Riesz operator defined on a Banach space is a finite rank operator. 2010 Mathematics Subject Classification. 47B06, 46L05. 1. Introduction. Let X be a Banach space, and denote by L(X) the Banach algebra of all bounded linear operators on X. An operator T ∈ L(X) is called a Riesz operator if the coset T + K(X) is quasinilpotent in the quotient algebra L(X)/K(X), where K(X) is the closed ideal of compact operators in L(X). We refer the reader to Dowson ((4), Part 2) for some basic properties of Riesz operators. For T ∈ L(X), denote the null space of T by N(T) and the range of T by R(T). The smallest integer n such that N(T n ) = N(T n+1 ) is called the ascent of T and it is denoted by α(T). The descent of T is the smallest integer n such that R(T n ) = R(T n+1 ) and it is denoted by δ(T). If M is a closed subspace of X invariant under T (i.e. T(M) ⊆ M), then the operator TM defined in L(X/M )b yTM(x + M) = Tx+ M is called the induced operator of T by M. The restriction of T to M is denoted by T|M.I fA is a C ∗ -algebra then the operator

Radius preserving (semi)regularities in Banach algebras

Abstract In this note we investigate regularities (semiregularities) R and S in a Banach algebra A satisfying S ⊂ R and the corresponding spectra σS and σR Satisfying sup {|λ| : λ ∈ σR(a)} = sup{|λ| : λ ∈ σS (a)} for all a ∈ A.