Journal of Operator Theory

Volume 36, Issue 2, Fall 1996 pp. 249-281.

A spectral classification of operators related to polynomial boundedness

Authors: Rolf Gohm
Author institution:Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, GERMANY

Summary: A local version of the concept of polynomial boundedness for operators on Banach spaces is defined and its relations to functional calculi are examined. For certain positive operators on $L^\infty$-spaces, especially for endomorphisms, lack of local polynomial boundedness corresponds to mixing properties. In particular, we give a new characterization of the weak mixing property. Some results extend to more general C*-algebras. This is done by constructing certain topological embeddings of the unit vector base of $l^1(\mathbb N_0)$ into the orbits of an operator. To analyze the underlying structure we introduce the concept of a transition set. We compute transition sets for the shift operator on $l^1(\mathbb Z)$ and show how to define a corresponding similarity invariant.

Keywords: Polynomial boundedness, functional calculi, ergodic theory, embeddings of sequence spaces, similarity.

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