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On nilpotent operators

Laura Burlando — 2005

Studia Mathematica

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 ( ε ) n 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

Laura Burlando — 2004

Studia Mathematica

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 n - 2 T converges uniformly to zero, the sum of the range and the kernel of 1-T being closed, and n - 3 k = 0 n - 1 A k converges uniformly, the sum of the range of 1-A and the kernel of (1-A)² being...

The closure of the invertibles in a von Neumann algebra

Laura BurlandoRobin Harte — 1996

Colloquium Mathematicae

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...

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