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Bounded point evaluations for multicyclic operators. (English) Zbl 1079.47009

Jarosz, Krzysztof (ed.) et al., Topological algebras, their applications, and related topics. Proceedings of the conference to celebrate the 70th birthday of Professor Wiesław Żelazko, Bȩdlewo, Poland, May 11–17, 2003. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 67, 199-217 (2005).
An operator \(T\) acting on a Banach space \(X\) is said to be \(n\)-multicyclic, where \(n\) is a positive integer, if there are \( x_i\), \(1\leq i \leq n\), in \(X\) such that \(X=\overline {\text{span}}\, \{T^mx_i : 1\leq i \leq n \text{ and } m\geq 0\}\) and \(n\) is the smallest positive integer with this property. For \(n=1\), the operator \(T\) is said to be cyclic.
In the paper under review, some of the authors’ previous results about cyclic operators [see M. Mbekhta and E. H. Zerouali, J. Funct. Anal. 206, 69–86 (2004; Zbl 1053.47004)] are extended to multicyclic operators. Among several interesting results relating different spectral quantities and bounded point evaluations, the following one must be emphasized: for a multicyclic operator \(T\) satisfying the Bishop property \((\beta) \), the set of analytic point evaluations is exactly equal to the set of bounded point evaluations minus the approximate spectrum. One of the keys to prove the latter result is a new notion of analytic structure, which is different from the one introduced by Herrero [see D. A. Herrero and L. Rodman, Indiana Univ. Math. J. 34, 619–629 (1985; Zbl 0574.47003)]. In particular, the connections between both analytic structures and bounded point evaluations are analyzed.
The authors also apply their results to extend or to give simpler proofs of previous results. A very elegant one is that of a theorem of D. Herrero and L. Rodman that asserts that the set of cyclic \(n\)-tuples for \(T\) is dense in the inflation \(X^{(n)}\) if and only if the the set of analytic point evaluations is empty.
For the entire collection see [Zbl 1063.46001].

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
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