[Banach] Abstract of a paper by Petr Hajek and Richard J. Smith

This is an announcement for the paper "Operator machines on directed graphs" by Petr Hajek and Richard J. Smith.

Abstract: We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X --> X such that the set A = {x in X : ||R^n(x)|| --> infinity} is non-empty and nowhere dense in X. Moreover, if x in X\A then some subsequence of (R^n(x)) converges weakly to x. This answers in the negative a recent conjecture of Prajitura. The result can be extended to any Banach space containing an infinite-dimensional, complemented subspace with a symmetric basis; in particular, all 'classical' Banach spaces admit such an operator.

Archive classification: math.FA

Mathematics Subject Classification: 47A05

The source file(s), machines14.tex: 47356 bytes, is(are) stored in gzipped form as 0906.0160.gz with size 14kb. The corresponding postcript file has gzipped size 111kb.

Submitted from: smith@math.cas.cz

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