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Geometry of the spectral semidistance in Banach algebras

Gareth Braatvedt; Rudi Brits — 2014

Czechoslovak Mathematical Journal

Let $A$ be a unital Banach algebra over $ℂ$ , and suppose that the nonzero spectral values of $a$ and $b \in A$ are discrete sets which cluster at $0 \in ℂ$ , 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, we further show that $a$ and $b$ are quasinilpotent equivalent if...

Rank, trace and determinant in Banach algebras: generalized Frobenius and Sylvester theorems

Gareth Braatvedt; Rudolf Brits; Francois Schulz — 2015

Studia Mathematica

As a follow-up to a paper of Aupetit and Mouton (1996), we consider the spectral definitions of rank, trace and determinant applied to elements in a general Banach algebra. We prove a generalization of Sylvester's Determinant Theorem to Banach algebras and thereafter a generalization of the Frobenius inequality.

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