×

\(m\)-isometries on Banach spaces. (English) Zbl 1230.47018

Summary: We introduce the notion of an \(m\)-isometry of a Banach space, following a definition of J. Agler and M. Stankus [Integral Equations Oper. Theory 21, No. 4, 383–429 (1995; Zbl 0836.47008)] in the Hilbert space setting. We give a first approach to the general theory of these maps. Then, we focus on the dynamics of \(m\)-isometries, showing that they are never \(N\)-supercyclic. This result is new even on a Hilbert space, and even for isometries on a general Banach space.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators

Citations:

Zbl 0836.47008
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agler, m-isometric transformations of Hilbert space. I, Integral Equations Oper. Theory 21 pp 383– (1995) · Zbl 0836.47008
[2] Agler, m-isometric transformations of Hilbert space. II, Integral Equations Oper. Theory 23 pp 1– (1995) · Zbl 0857.47011
[3] Agler, m-isometric transformations of Hilbert space. III, Integral Equations Oper. Theory 24 pp 379– (1996) · Zbl 0871.47012
[4] Ansari, Some properties of cyclic operators, Acta Sci. Math. 63 pp 195– (1997) · Zbl 0892.47004
[5] Bayart, Cambridge Tracts in Mathematics Vol. 179 (2009)
[6] T. Bermúdez I. Marrero A. Martinón On the orbit of an m -isometry 2009
[7] Bourdon, Some properties of n-supercyclic operators, Stud. Math. 165 pp 135– (2004) · Zbl 1056.47008
[8] Feldman, N-supercyclic operators, Stud. Math. 151 pp 141– (2002) · Zbl 1006.47008
[9] Kérchy, Hyperinvariant subspaces of operators with non-vanishing orbits, Proc. Am. Math. Soc. 127 pp 1363– (1999) · Zbl 0914.47004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.