Power bounded operators and supercyclic vectors

Müller, V.

doi:10.1090/S0002-9939-03-06962-4

Power bounded operators and supercyclic vectors
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Proc. Amer. Math. Soc. 131 (2003), 3807-3812 Request permission

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, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant positive cone.

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