Vol. 68, No. 1, 1977

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Absolutely divergent series and isomorphism of subspaces. II

William Henry Ruckle

Vol. 68 (1977), No. 1, 229–240

The following relations between a Banach space E and a Banach space X are, roughly speaking, generalizations of the relation “E is a closed subspace of X.”

(LIX) The finite dimensional subspaces of E are uniformly isomorphic to subspaces of X under omorphisms which extend to all of E without increase of norm.

(SpX) Finite rank mappings from any Banach space into E can be uniformly factored through subspaces of X.

(ASX) The continuous linear mappings from E into X distinguish the absolutely summing mappings Srom any Banach space into E.

(SIX) For each absolutely divergent series nxn in E there is a continuous linear mapping T from E into X such that nTxn diverges absolutely.

Our main result is that these four conditions are equivalent if X contains a subspace isomorphic to λ[X] where λ is a normal BK-space. A related result of some interest is that the class of continuous linear mappings which factor through spaces which contain a complemented copy of λ[X] forms a Banach operator ideal.

Mathematical Subject Classification 2000
Primary: 46B99
Received: 14 November 1975
Published: 1 January 1977
William Henry Ruckle