Geometry of the Spectral Semidistance in Banach Algebras

TL;DR

This paper introduces a geometric formula for the spectral semidistance in Banach algebras based on spectra and Riesz projections, extending previous results and simplifying computations for certain operator classes.

Contribution

It develops a new geometric formula for spectral semidistance in Banach algebras that depends on spectra and Riesz projections, extending prior work and simplifying calculations.

Findings

The formula depends on spectra and Riesz projections.

A characterization of quasinilpotent equivalence via Riesz projections.

Simplification for decomposable operators like compact and Riesz operators.

Abstract

Let $A$ be a unital Banach algebra over $\mathbb C$, and suppose that the nonzero spectral values of, respectively, $a\mbox{ and }b\in A$ are discrete sets which cluster at $0\in\mathbb C$, if anywhere. We develop a plane geometric formula for the spectral semidistance of $a$ and $b$ which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, it is further shown that $a$ and $b$ are quasinilpotent equivalent if and only if all the Riesz projections, $p(\alpha,a)$ and $p(\alpha,b)$, correspond. For certain important classes of decomposable operators (compact, Riesz, etc.) the proposed formula reduces the involvement of the underlying Banach space $X$ in the computation of the spectral semidistance, and appears to be a useful alternative to Vasilescu's geometric formula (which requires knowledge of the local spectra of the operators at each $0\not=x\in X$). The apparent advantage gained through the use of a global spectral parameter in the formula aside, the various methods of complex analysis could then be employed to deal with the spectral projections; we give examples illustrating the utility of the main results.