Geometry of the Spectral Semidistance in Banach Algebras

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.