Multiplicity of direct sums of operators on Banach spaces | Zendy

Grivaux Sophie | Zendy; Roginskaya Maria | Zendy

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Multiplicity of direct sums of operators on Banach spaces

Author(s) -

Grivaux Sophie,

Roginskaya Maria

Publication year - 2009

Publication title -

bulletin of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 2.396

H-Index - 48

eISSN - 1469-2120

pISSN - 0024-6093

DOI - 10.1112/blms/bdp084

Subject(s) - mathematics , multiplicity (mathematics) , banach space , sequence (biology) , hilbert space , bounded function , operator (biology) , regular polygon , combinatorics , compact operator , integer (computer science) , discrete mathematics , operator space , pure mathematics , finite rank operator , mathematical analysis , biochemistry , chemistry , genetics , geometry , repressor , gene , transcription factor , computer science , extension (predicate logic) , biology , programming language

Let T be a bounded operator on a complex Banach space X and let T n 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 ( T n )) n ⩾ 1 of the multiplicities of the operators T n . 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 ( T n ) = 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 ( T n )) n ⩾ 0 for some operator T on a Hilbert space.

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