Power bounded operators and supercyclic vectors II

Müller, V.

doi:10.1090/S0002-9939-05-07829-9

Power bounded operators and supercyclic vectors II
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Proc. Amer. Math. Soc. 133 (2005), 2997-3004 Request permission

Abstract:

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. For non-reflexive Banach spaces these results remain true; however, the non-supercyclic vector (invariant cone, respectively) relates to the adjoint of the operator.

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