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HB ‐subspaces and Godun sets of subspaces in Banach spaces
Author(s) -
Oja Eve
Publication year - 1997
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s002557930001202x
Subject(s) - mathematics , banach space , subspace topology , linear subspace , ideal (ethics) , generalization , combinatorics , discrete mathematics , operator (biology) , space (punctuation) , invariant subspace problem , approximation property , pure mathematics , mathematical analysis , unbounded operator , philosophy , biochemistry , chemistry , linguistics , epistemology , repressor , transcription factor , gene
Let X be a Banach space and Y its closed subspace having property U in X . We use a net ( A α) of continuous linear operators on X such that ‖ A α ‖ ≤ 1, A α ( X ) ⊂ Y for all α, and lim α g ( A α y ) = g ( y ), y ∈ Y , g ∈ Y * to obtain equivalent conditions for Y to be an HB ‐subspace, u ‐ideal or h ‐ideal of X . Some equivalent renormings of c 0 and l 2 are shown to provide examples of spaces X for which K ( X ) has property U in L ( X ) without being an HB ‐subspace. Considering a generalization of the Godun set [3], we establish some relations between Godun sets of Banach spaces and related operator spaces. This enables us to prove e.g ., that if K ( X ) is an HB ‐subspace of L ( X ), then X is an HB ‐subspace of X ** —the result conjectured to be true by Å. Lima [9].