Characterizing Fréchet–Schwartz spaces via power bounded operators

Power bounded weighted composition operators

We study when weighted composition operators Cφ,ψ acting between weighted Bergman spaces of infinite order are power bounded resp. uniformly mean ergodic.

Power - Bounded Operators and Relatednorm

Power Bounded Operators and Supercyclic Vectors

Abstract. By the well-known result of Brown, Chevreau and Pearcy, every Hilbert space contraction with spectrum containing the unit circle has a nontrivial closed invariant subspace. Equivalently, there is a nonzero vector which is not cyclic. We show that each power bounded operator on a Hilbert space with spectral radius equal to one has a nonzero vector which is not supercyclic. Equivalently...

Power Bounded Operators and Supercyclic Vectors Ii

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...

Power-bounded Operators and Related Norm Estimates

It is considered whether L= lim supn→∞ n‖Tn+1 − Tn‖<∞ implies that the operator T is power-bounded. It is shown that this is so if L < 1/e, but it does not necessarily hold if L=1/e. As part of the methods, a result of Esterle is improved, showing that if σ(T )= {1} and T = I, then lim infn→∞ n‖Tn+1 − Tn‖ 1/e. The constant 1/e is sharp. Finally, a way to create many generalizations of Esterle’s...