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Operator Machines on Directed Graphs

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    0352530 - MÚ 2011 RIV CH eng J - Journal Article
    Hájek, Petr Pavel - Smith, R.J.
    Operator Machines on Directed Graphs.
    Integral Equations and Operator Theory. Roč. 67, č. 1 (2010), s. 15-31. ISSN 0378-620X. E-ISSN 1420-8989
    R&D Projects: GA AV ČR IAA100190801
    Institutional research plan: CEZ:AV0Z10190503
    Keywords : orbits of operators
    Subject RIV: BA - General Mathematics
    Impact factor: 0.521, year: 2010
    http://link.springer.com/article/10.1007%2Fs00020-010-1766-y

    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 is an element of X : parallel to R(n)x parallel to -> infinity} is non-empty and nowhere norm-dense in X. Moreover, if x is an element of X/A then some subsequence of (R-n x)(n=1)(infinity) 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.
    Permanent Link: http://hdl.handle.net/11104/0192020

     
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