# On decompositions of Banach spaces into a sum of operator ranges

Studia Mathematica (1999)

• Volume: 132, Issue: 1, page 91-100
• ISSN: 0039-3223

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## Abstract

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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 $X={A}_{1}\left({Y}_{1}\right)+{A}_{2}\left({Y}_{2}\right)$ 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 $X={T}_{1}\left({Z}_{1}\right)+{T}_{2}\left({Z}_{2}\right)$ such that neither of the operator ranges ${T}_{1}\left({Z}_{1}\right)$, ${T}_{2}\left({Z}_{2}\right)$ contains an infinite-dimensional closed subspace if and only if X does not contain an isomorphic copy of ${l}_{1}$.

## How to cite

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Fonf, V., and Shevchik, V.. "On decompositions of Banach spaces into a sum of operator ranges." Studia Mathematica 132.1 (1999): 91-100. <http://eudml.org/doc/216588>.

@article{Fonf1999,
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 $l_1$.},
author = {Fonf, V., Shevchik, V.},
journal = {Studia Mathematica},
keywords = {decomposition of Banach spaces; representation; infinite-codimensional operator ranges},
language = {eng},
number = {1},
pages = {91-100},
title = {On decompositions of Banach spaces into a sum of operator ranges},
url = {http://eudml.org/doc/216588},
volume = {132},
year = {1999},
}

TY - JOUR
AU - Fonf, V.
AU - Shevchik, V.
TI - On decompositions of Banach spaces into a sum of operator ranges
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 1
SP - 91
EP - 100
AB - 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 $l_1$.
LA - eng
KW - decomposition of Banach spaces; representation; infinite-codimensional operator ranges
UR - http://eudml.org/doc/216588
ER -

## References

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1. [1] Fillmore P. A., and Williams J. P., On operator ranges, Adv. Math. 7 (1971), 254-281. Zbl0224.47009
2. [2] Fonf V. P., On supportless absorbing convex subsets in normed spaces, Studia Math. 104 (1993), 279-284. Zbl0815.46019
3. [3] Fonf V. P., and Shevchik V. V., Representing a Banach space as a sum of operator ranges, Funct. Anal. Appl. 29 (1995), no. 3, 220-221 (transl. from the Russian). Zbl0867.47004
4. [4] Gowers W. T., and Maurey B., The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851-874. Zbl0827.46008
5. [5] Johnson W. B., and Rosenthal H. P., On w*-basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77-92. Zbl0213.39301
6. [6] Lindenstrauss J., and Tzafriri L., Classical Banach Spaces, Vol. 1, Springer, Berlin, 1977. Zbl0362.46013
7. [7] Pełczyński A., On strictly singular and strictly cosingular operators, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 13 (1965), 31-41. Zbl0138.38604
8. [8] Pietsch A., Operator Ideals, North-Holland, Amsterdam, 1980.

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