Best approximation in spaces of compact operators

• Published in: Linear algebra and its applications Vol. 627; pp. 72 - 79
• Main Author: Rao, T.S.S.R.K.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.10.2021; American Elsevier Company, Inc
• Subjects: [formula omitted]-predual spaces; Approximation; Banach spaces; Compact operators; Injective and projective tensor products; Linear algebra; Linear operators; Mathematical analysis; Strongly proximinal subspaces; Subspaces; Strongly proximinal subspaces; Compact operators; L1-predual spaces; Injective and projective tensor products; primary
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2021.06.006

Let K(X,Y) be the space of compact operators. For a proximinal subspace Z⊂Y, this paper deals with the question, when does every Y-valued compact operator admit a Z-valued compact best approximation? For any reflexive Banach space X and for a L1-predual space Y, if Z⊂Y is a strongly proximinal subspace of finite codimension, we show that K(X,Z) is a proximinal subspace of K(X,Y) under an additional condition on the position of K(X,Z). When Y is a c0-direct sum of finite dimensional spaces we achieve a strong transitivity result by showing that for any proximinal subspace of finite codimension Z⊂Y, every Y-valued bounded operator admits a best Z-valued compact approximation.