Equal signs additive sequences in Banach spaces |
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Authors: | Antoine Brunel Louis Sucheston |
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Institution: | 1. University of Paris VI, 9, Quai St. Bernard, Tour 56, 75005 Paris France;2. Department of Mathematics, Ohio State University, Columbus, Ohio 43210 U.S.A. |
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Abstract: | A sequence (en) spanning a Banach space E is called ESA or equal signs additive if the norm of a linear combination of the ei's does not change when adjacent coefficients of equal sign are combined. Call the sequence (en) regular if neither E nor its dual contain an isomorphic copy of c0. It is shown that a regular ESA sequence is a boundedly complete and 1-shrinking basis for its span which is thus quasi-reflexive. It is further possible to replace a regular ESA norm by an equivalent ESA norm rendering E isometrically isomorphic to its second dual. A sequence (en) is called IS or invariant under spreading if the norm of a linear combination of the ei's does not change when the mutual distances of the terms in the sequence (but not their relative positions) change. We give a simple construction of an unconditional norm for an IS sequence, hence, in particular, for an ESA sequence. Also, an inverse construction is obtained: We prove that each unconditional IS basis gives rise to an ESA basis by means of an inversion formula; to nonequivalent IS unconditional bases correspond nonequivalent ESA bases. It follows that nonisomorphic ESA bases are plentiful. |
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