On decompositions of Banach spaces into a sum of operator ranges

V. P. Fonf; V. Shevchik

ArticleOriginal scientific text

Title

On decompositions of Banach spaces into a sum of operator ranges

Authors ¹, ²

Affiliations

Department of Mathematics and Computer Sciences, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel
Department of Mathematics and Informatics, Friedrich Schiller University, Lentragraben 1, 07743 Jena, Germany

Abstract

It is proved that a separable Banach space X admits a representation

X = X_{1} + X_{2}

as a sum (not necessarily direct) of two infinite-codimensional closed subspaces

X_{1}

and

X_{2}

if and only if it admits a representation

X = A_{1} (Y_{1}) + A_{2} (Y_{2})

as a sum (not necessarily direct) of two infinite-codimensional operator ranges. Suppose that a separable Banach space X admits a representation as above. Then it admits a representation

X = T_{1} (Z_{1}) + T_{2} (Z_{2})

such that neither of the operator ranges

T_{1} (Z_{1})

,

T_{2} (Z_{2})

contains an infinite-dimensional closed subspace if and only if X does not contain an isomorphic copy of

l_{1}

.

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Title

On decompositions of Banach spaces into a sum of operator ranges

Affiliations

Abstract

Bibliography