An Approximation Problem in the Space of Bounded Operators. (25th January 2023)

• Record Type: Journal Article
• Authors: Mal, Arpita
• Is Part Of: Numerical functional analysis and optimization. Volume 44:Number 2(2023)
• Page Start: 124
• Page End: 137
• Publication Date: 2023-01-25
• Subjects: 46B25 -- 47L05 -- Primary 46B28 -- Secondary 41A50; Best approximation -- Birkhoff–James orthogonality -- distance formulae -- linear operators -- L1− predual; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2022.2158197
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: For Banach spaces X, Y, we consider a distance problem in the space of bounded linear operators L ( X, Y ) . Motivated by a recent article, we obtain sufficient conditions so that for a compact operator T ∈ L ( X, Y ) and a closed subspace Z ⊂ Y, the following equation holds, which relates global approximation with local approximation: d ( T, L ( X, Z ) ) = sup { d ( T x, Z ) : x ∈ X, ‖ x ‖ = 1 } . In some cases, we show that the supremum is attained at an extreme point of the corresponding unit ball. Furthermore, we obtain some situations when the following equivalence holds: T ⊥ B L ( X, Z ) ⇔ T * * x 0 * * ⊥ B Z ⊥ ⊥ ⇔ T * * ⊥ B L ( X * *, Z ⊥ ⊥ ), for some x 0 * * ∈ X * * satisfying ‖ T * * x 0 * * ‖ = ‖ T * * ‖ ‖ x 0 * * ‖, where Z ⊥ is the annihilator of Z . One such situation is when Z is an L 1 − predual space and an M − ideal in Y and T is a multi-smooth operator of finite order. Another such situation is when X is an abstract L 1 − space and T is a multi-smooth operator of finite order. Finally, as a consequence of the results, we obtain a sufficient condition for proximinality of a subspace Z in Y .