Teresa Bermúdez

On hypercyclicity and supercyclicity criteria

We show that the Hypercyclicity Criterion coincides with other existing hypercyclicity criteria and prove that a wide class of hypercyclic operators satisfy the Criterion. The results obtained extend or improve earlier work of several authors. We also unify the different versions of the Supercyclicity Criterion and show that operators with dense generalised kernel and dense range are supercyclic.

On the existence of chaotic and hypercyclic semigroups on banach spaces

We prove that every separable infinite dimensional complex Banach space admits a hypercyclic uniformly continuous semigroup. We also prove that there exist Banach spaces admitting no chaotic strongly continuous semigroups.

The range of operators on von Neumann algebras

We prove that for every bounded linear operator T: X → X, where X is a non-reflexive quotient of a von Neumann algebra, the point spectrum of T * is non-empty (i.e., for some A ∈ C the operator λI - T fails to have dense range). In particular, and as an application, we obtain that such a space cannot support a topologically transitive operator.

Perturbation of -Isometries by Nilpotent Operators

We prove that if is an -isometry on a Hilbert space and an -nilpotent operator commuting with , then is a -isometry. Moreover, we show that a similar result for -isometries on Banach spaces is not true.

On operatorsT such thatf(T) is hypercyclic

We give conditions such that an operator given by the Dunford-Taylor functional calculus is supercyclic or hypercyclic. Indeed, we improve [15, Theorem 1].

On the boundedness of the local resolvent function

AbstractFor a hyponormal operatorT on a complex Hilbert spaceH, we show that if the spectrum ofT has empty interior, then the local resolvent function,

Changes of signs in conditionally convergent series on a small set

\hat x_T

Li–Yorke and distributionally chaotic operators

, is unbounded for everyx∈H{0}. In particular, ifT is selfadjoint, then