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Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces

Teresa BermúdezAntonio BonillaJosé A. ConejeroAlfredo Peris — 2005

Studia Mathematica

Our aim in this paper is to prove that every separable infinite-dimensional complex Banach space admits a topologically mixing holomorphic uniformly continuous semigroup and to characterize the mixing property for semigroups of operators. A concrete characterization of being topologically mixing for the translation semigroup on weighted spaces of functions is also given. Moreover, we prove that there exists a commutative algebra of operators containing both a chaotic operator and an operator which...

Powers of m-isometries

Teresa BermúdezCarlos Díaz MendozaAntonio Martinón — 2012

Studia Mathematica

A bounded linear operator T on a Banach space X is called an (m,p)-isometry for a positive integer m and a real number p ≥ 1 if, for any vector x ∈ X, k = 0 m ( - 1 ) k ( m k ) | | T k x | | p = 0 . We prove that any power of an (m,p)-isometry is also an (m,p)-isometry. In general the converse is not true. However, we prove that if T r and T r + 1 are (m,p)-isometries for a positive integer r, then T is an (m,p)-isometry. More precisely, if T r is an (m,p)-isometry and T s is an (l,p)-isometry, then T t is an (h,p)-isometry, where t = gcd(r,s) and h = min(m,l)....

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