# 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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topFonf, 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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- [8] Pietsch A., Operator Ideals, North-Holland, Amsterdam, 1980.

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