On decompositions of Banach spaces into a sum of operator ranges

V. P. Fonf; V. V. Shevchik

doi:10.4064/sm-132-1-91-100

On decompositions of Banach spaces into a sum of operator ranges

V. P. Fonf, V. V. Shevchik

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

It is proved that a separable Banach space X admits a representation X = X₁ + X₂ as a sum (not necessarily direct) of two infinite-codimensional closed subspaces X₁ and X₂ if and only if it admits a representation X = A₁(Y₁) + A₂(Y₂) 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 = T₁(Z₁) + T₂(Z₂) such that neither of the operator ranges T₁(Z₁), T₂(Z₂) contains an infinite-dimensional closed subspace if and only if X does not contain an isomorphic copy of l₁.

Original language	English
Pages (from-to)	91-100
Number of pages	10
Journal	Studia Mathematica
Volume	132
Issue number	1
DOIs	https://doi.org/10.4064/sm-132-1-91-100
State	Published - 1 Jan 1999

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Mathematics (all)

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N2 - It is proved that a separable Banach space X admits a representation X = X1 + X2 as a sum (not necessarily direct) of two infinite-codimensional closed subspaces X1 and X2 if and only if it admits a representation X = A1(Y1) + A2(Y2) 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 = T1(Z1) + T2(Z2) such that neither of the operator ranges T1(Z1), T2(Z2) contains an infinite-dimensional closed subspace if and only if X does not contain an isomorphic copy of l1.

AB - It is proved that a separable Banach space X admits a representation X = X1 + X2 as a sum (not necessarily direct) of two infinite-codimensional closed subspaces X1 and X2 if and only if it admits a representation X = A1(Y1) + A2(Y2) 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 = T1(Z1) + T2(Z2) such that neither of the operator ranges T1(Z1), T2(Z2) contains an infinite-dimensional closed subspace if and only if X does not contain an isomorphic copy of l1.

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On decompositions of Banach spaces into a sum of operator ranges

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