Generalized spectral theory in complex Banach algebras (original) (raw)

The relative spectrum of elements in a real Banach algebra

Mathematische Annalen, 1980

In dealing with the spectrum of an element in a real Banach algebra it is necessary to complexify the Banach algebra in order to get meaningful results. This procedure, at first, may not seem much of a sacrifice. But it does lead to some difficulties when we use the resoIvent of an element in the original algebra to generate idempotents in the original algebra. The following is an attempt to get around this problem and hopefully lead to a more full theory for real Banach algebras and a useful generalization of the concept of spectrum. 1. Definitions and Basic Notions will denote a real Banach algebra with identity, e. Definition t. Let be~ and ms~. The spectrum of b relative to m, called the mspectrum of b, is the subset of Nz defined by spin(b)= {(x,y)~ z :b-(xe+ym) is not invertible in ~}, The m-resolvent of b is defined as the complement in N2 of sp,,(b), that is res,,(b) = {(x, y)~ N ~ : [b-(xe + ym)]-i exists in .~}. Note that if.~ is a complex Banach algebra (and hence also a real algebra) and m = i, then the "/-spectrum" and "i-resotvent" of b are just the ordinary complex spectrum and complex resolven.t of b. When we specialize our theory to complex algebras and speak of the "ordinary spectrum", sp(b), of an element b we will mean this complex spectrum. If ~ is a real algebra but not already a complex algebra, by the spectrum of b we mean the complex spectrum of b in the larger complex algebra Ne. obtained by complexifying N [4]. We shall also speak, in either real or complex algebras, of the "real spectrum" of an element b, by which we mean the collection of real numbers x such that b-xe is not invertible. As is welt known from the Gelfand theory (see [4], Theorem(3.1.6)), for commutative Banach algebras we have sp(b)= {cr(b):a is a nonzero homomorphism from ~ into 112} and real spectrum (b)= lRc~sp(b).

The spectra of some algebras of analytic mappings

Indagationes Mathematicae, 1999

Let E be a Banach space with the approximation property and let F be a Banach algebra with identity. We study the spectrum of the algebra '&,(E, F) of all holomorphic mappings f : E -F that are bounded on the bounded subsets of E.

Finite spectra and quasinilpotent equivalence in Banach algebras

Czechoslovak Mathematical Journal, 2012

This paper further investigates the implications of quasinilpotent equivalence for (pairs of) elements belonging to the socle of a semisimple Banach algebra. Specifically, not only does quasinilpotent equivalence of two socle elements imply spectral equality, but also the trace, determinant and spectral multiplicities of the elements must agree. It is hence shown that quasinilpotent equivalence is established by a weaker formula (than that of the spectral semidistance). More generally, in the second part, we show that two elements possessing finite spectra are quasinilpotent equivalent if and only if they share the same set of Riesz projections. This is then used to obtain further characterizations in a number of general, as well as more specific situations. Thirdly, we show that the ideas in the preceding sections turn out to be useful in the case of C * -algebras, but now for elements with infinite spectra; we give two results which may indicate a direction for further research.

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.

On generalized spectra in C*-algebras of operators over C*-algebras

arXiv (Cornell University), 2020

In this paper we consider shift operators, self-adjoint, unitary and normal operators on the standard module over a unital C *-algebra A. We define various generalized spectra in A of these operators, give description of such spectra of these operators and investigate their further properties.

Spectral characterizations of scalars in a Banach algebra

Bulletin of the London Mathematical Society, 2009

For a complex Banach algebra A with unit 1, we give several characterizations of the scalars, that is, multiples of the identity. To a large extent, this work is a continuation and generalization of the work done on characterizations of the radical in Banach algebras. In particular it is shown that if a ∈ A has the property that the number of elements in the spectrum of ax is less than or equal to the number of elements in the spectrum of x for all x in an arbitrary neighbourhood of 1, then a is a scalar. Moreover, as a consequence of some of the results, new spectral characterizations of commutative Banach algebras are obtained. In particular, A is commutative if and only if it has the property that the number of elements in the spectrum remains invariant under all permutations of three elements in some neighbourhood of the identity.

Properties of the Drazin spectra for Banach space operators and Banach algebra elements

Given a Banach Algebra AAA and ainAa\in AainA, several relationships among the Drazin spectrum of aaa and the ascent, the descent and the Drazin spectra of the multiplication operators LaL_aLa and RaR_aRa will be presented; the Banach space operator case will be also examined. In addition, a characterization of the spectrum of aaa in terms of the Drazin spectrum and the poles of the resolvent of aaa will be considered. Furthermore, several basic properties of the Drazin spectrum in Banach algebras will be studied.

On Real and Complex Spectra in some Real C*-Algebras and Applications

Zeitschrift für Analysis und ihre Anwendungen, 1999

A real extension A of a complex C-algebra A by some element rn which has a number of special properties is proposed. These properties allow us to introduce some suitable operations of addition, multiplication and involution on A. After then we are able to study Moore-Penrose invertibility in A. Because this notion strongly depends on the element m, we study under what conditions different elements m produce just the same involution on A. It is shown that the set of all additive continuous operators £aa(fl) acting in a complex Hilbert space fl possesses unique involution only (in the sense defined below). In addition, we consider some properties of the real and complex spectra of elements belonging to A, and show that whenever an operator sequence {A,} C £aa(fl) is weakly asymptotically Moore-Penrose invertible, then the real spectrum of AA can be split in two special parts. This property has been earlier known for sequences of linear operators.