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.

/pdf/ruston-elements-and-fredholm-theory-relative-to-arbitrary-51nctchjgo.pdf

Ruston elements and fredholm theory relative to arbitrary homomorphisms

In this note we study the behaviour of the spectrum and exponential spectrum under perturbation by inessential and Riesz elements.

Perturbation by inessential and riesz elements

Different exponential spectra in Banach algebras

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 Satisfyingsup {|λ| : λ ∈ σR(a)} = sup{|λ| ...

Radius preserving (semi)regularities in Banach algebras

By considering arbitrary mappings ω from a Banach algebra A into the set of all nonempty, compact subsets of the complex plane such that for all a ∈ A, the set ω(a) lies between the boundary and connected hull of the exponential spectrum of a, we create a general framework in which to generalize a number of results involving spectra such as the exponential and singular spectra. In particular, we discover a number of new properties of the boundary spectrum.

Generalized Spectral Perturbation and the Boundary Spectrum

Let A be a unitary Banach algebra and let B be a closed subalgebra of A that contains the unit. For an element of B we show in this note how both the singular and exponential spectrum depends on the algebra.

A note on the singular and exponential spectrum in banach algebras

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A note on the singular and exponential spectrum in banach algebras