On decompositions of Banach spaces into a sum of operator ranges

Fonf, V.; Shevchik, V.

Fonf, V. ; Shevchik, V.

Studia Mathematica, Tome 133 (1999), p. 91-100 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $l_{1}$ .

Publié le : 1999-01-01

Zbl 0934.46016

EUDML-ID : urn:eudml:doc:216588

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     title = {On decompositions of Banach spaces into a sum of operator ranges},
     journal = {Studia Mathematica},
     volume = {133},
     year = {1999},
     pages = {91-100},
     zbl = {0934.46016},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv132i1p91bwm}
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Fonf, V.; Shevchik, V. On decompositions of Banach spaces into a sum of operator ranges. Studia Mathematica, Tome 133 (1999) pp. 91-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv132i1p91bwm/

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