On rank one and finite elements of Banach algebras
We give a spectral characterisation of rank one elements and of the socle of a semisimple Banach algebra.
On regularities and Fredholm theory
We investigate the relationship between the regularities and the Fredholm theory in a Banach algebra.
The index for Fredholm elements in a Banach algebra via a trace
We show that the existence of a trace on an ideal in a Banach algebra provides an elegant way to develop the abstract index theory of Fredholm elements in the algebra. We prove some results on the problem of the equality of the nonzero exponential spectra of elements ab and ba and use the index theory to formulate a condition guaranteeing this equality in a quotient algebra.
A note on the singular spectrum.
We compare the singular spectrum of a Banach algebra element with the usual spectrum and exponential spectrum.
Rank and the Drazin inverse in Banach algebras
Let A be an arbitrary, unital and semisimple Banach algebra with nonzero socle. We investigate the relationship between the spectral rank (defined by B. Aupetit and H. Mouton) and the Drazin index for elements belonging to the socle of A. In particular, we show that the results for the finite-dimensional case can be extended to the (infinite-dimensional) socle of A.
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