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Operators Birkhoff–James Orthogonal to Spaces of Operators.
- Source :
- Numerical Functional Analysis & Optimization; 2021, Vol. 42 Issue 10, p1201-1208, 8p
- Publication Year :
- 2021
-
Abstract
- Let L (X , Y) be the space of bounded linear operators between real Banach spaces X, Y. For a closed subspace Z ⊂ Y , we partially solve the operator version of Birkhoff–James orthogonality problem, if T ∈ L (X , Y) is orthogonal to L (X , Z) , when does there exist a unit vector x<subscript>0</subscript> such that | | T (x 0) | | = | | T | | and T (x 0) is orthogonal to Z? In order to achieve this we first develop a compact optimization for a Y-valued compact operator T, via a minimax formula for d (T , L (X , Z)) in terms of point-wise best approximations, that links local optimization and global optimization. Our result gives an operator version of a classical minimax formula of Light and Cheney, proved for continuous vector-valued functions. For any separable reflexive Banach space X and for Z ⊂ Y is a L<superscript>1</superscript>-predual space as well as a M-ideal in Y, we show that if T ∈ K (X , Y) is orthogonal to L (X , Z) , then there is a unit vector x<subscript>0</subscript> with | | T (x 0) | | = | | T | | and T (x 0) is orthogonal to Z. In the general case we also give some local conditions on T, when this can be achieved. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 01630563
- Volume :
- 42
- Issue :
- 10
- Database :
- Complementary Index
- Journal :
- Numerical Functional Analysis & Optimization
- Publication Type :
- Academic Journal
- Accession number :
- 153046183
- Full Text :
- https://doi.org/10.1080/01630563.2021.1952429