Multiplicity of direct sums of operators on Banach spaces

Journal article, 2009

Let T be a bounded operator on a complex Banach space X and let Tn be the direct sum T... T of n copies of T acting on X... X. The aim of this paper is to study the sequence (m(Tn))n>=1 of the multiplicities of the operators Tn. Answering a question of Atzmon, it is shown that this sequence is either eventually constant or grows to infinity at least as fast as n. Then examples of operators on Hilbert spaces, such that m(Tn) = d for every n>=1, are constructed, where d is an arbitrary positive integer. This answers a question of Herrero and Wogen and characterizes convex sequences that can be realized as a sequence (m(Tn))n 0 for some operator T on a Hilbert space.