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The Invariant Subspace Problem on Some Banach Spaces with Separable Dual
Author(s) -
Read C. J.
Publication year - 1989
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-58.3.583
Subject(s) - mathematics , invariant subspace problem , separable space , banach space , reflexive space , banach manifold , interpolation space , linear subspace , pure mathematics , approximation property , invariant (physics) , invariant subspace , reflexive operator algebra , lp space , subspace topology , eberlein–šmulian theorem , discrete mathematics , compact operator , mathematical analysis , functional analysis , extension (predicate logic) , computer science , mathematical physics , biochemistry , chemistry , gene , programming language
Operators without non‐trivial invariant subspaces (so called ‘cyclic operators’) are now known to exist on a number of Banach spaces, but all such Banach spaces are non‐reflexive and contain copies of the sequence space l 1 In this paper, we find cyclic operators on some new Banach spaces which do not contain l 1 . The Banach spaces involved are not reflexive, but they include the space c o , which has separable dual, and a space j ∞ (the l 2 direct sum of countably many copies of the James space J) which has a separable bidual (indeed, all the spacesJ ∞ * ,J ∞ ** ,J ∞ ***and so on, are separable). This seems to be a best possible result, short of finding a solution to the invariant subspace problem on a reflexive Banach space.