We show that each power bounded operator with spectral radius equal to one on a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant cone if 1 belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators. Fo...

In this short note, we answer a question of Martin and Sanders [Integr. Equ. Oper. Theory, 85 (2) (2016), 191-220] by showing the existence disjoint frequently hypercyclic operators which fail to be weakly mixing and, therefore, satisfy Disjoint Hypercyclicity Criterion. We also show that given an operator $T$ such $T \oplus T$ is hypercyclic, set $S$ $T, S$ are but Criterion SOT dense in algeb...

We give a sufficient condition for two operators to be disjointly frequently hypercyclic. apply this criterion composition acting on H(D) or the Hardy space H2(D). simplify result disjoint frequent hypercyclicity of pseudo shifts recent paper Martin et al. and we exhibit hypercyclic weighted shifts.