Continuity of spectrum and spectral radius in Banach algebras
This survey deals with necessary and/or sufficient conditions for continuity of the spectrum and spectral radius functions at a point of a Banach algebra.
A generalization of the uniform ergodic theorem to poles of arbitrary order
On nilpotent operators
We give several necessary and sufficient conditions in order that a bounded linear operator on a Banach space be nilpotent. We also discuss three necessary conditions for nilpotency. Furthermore, we construct an infinite family (in one-to-one correspondence with the square-summable sequences of strictly positive real numbers) of nonnilpotent quasinilpotent operators on an infinite-dimensional Hilbert space, all the iterates of each of which have closed range. Each of these operators (as well as...
Generalizations of Cesàro means and poles of the resolvent
An improvement of the generalization-obtained in a previous article [Bu1] by the author-of the uniform ergodic theorem to poles of arbitrary order is derived. In order to answer two natural questions suggested by this result, two examples are also given. Namely, two bounded linear operators T and A are constructed such that converges uniformly to zero, the sum of the range and the kernel of 1-T being closed, and converges uniformly, the sum of the range of 1-A and the kernel of (1-A)² being...
Continuity of spectrum and spectral radius in algebras of operators
On the problem of invariance under holomorphic functions for a set of continuity points of the spectrum function
The closure of the invertibles in a von Neumann algebra
In this paper we consider a subset  of a Banach algebra A (containing all elements of A which have a generalized inverse) and characterize membership in the closure of the invertibles for the elements of Â. Thus our result yields a characterization of the closure of the invertible group for all those Banach algebras A which satisfy  = A. In particular, we prove that  = A when A is a von Neumann algebra. We also derive from our characterization new proofs of previously known results, namely Feldman...
Comparisons between different spectra of an element in a Banach algebra.
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