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A Short Proof Concerning the Invariant Subspace Problem
Author(s) -
Read C. J.
Publication year - 1986
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-34.2.335
Subject(s) - counterexample , linear subspace , reflexive operator algebra , invariant subspace , mathematics , invariant subspace problem , invariant (physics) , subspace topology , unit sphere , operator (biology) , discrete mathematics , pure mathematics , banach space , combinatorics , compact operator , operator space , finite rank operator , mathematical analysis , computer science , mathematical physics , biochemistry , chemistry , repressor , transcription factor , gene , extension (predicate logic) , programming language
Counterexamples for the invariant subspace problem on a general Banach space exist due to Enflo [ 1 ], Read [ 3 ], Beauzamy [ 3 ] (simplification of [ 1 ]). On the space l 1 , there is a counterexample due to Read [ 4 ], which is rather long since it uses all of [ 3 ]. Here we present a short, direct proof that there is an invariant subspace free operator T on l 1 , and we also establish the following facts about our operator. First, it is a perturbation of a weighted shift operator by a nuclear operator. Secondly, its spectrum is identical with its approximate point spectrum, which is the unit ball of C. Thirdly, we can arrange either that no positive power of T has invariant subspaces, or alternately that every positive power T ∗ except T itself has non‐trivial invariant subspaces.