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An Approximation Problem in the Space of Bounded Operators

Mal, A (2022) An Approximation Problem in the Space of Bounded Operators. In: Numerical Functional Analysis and Optimization, 44 (2). pp. 124-137.

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Official URL: https://doi.org/10.1080/01630563.2022.2158197


For Banach spaces X, Y, we consider a distance problem in the space of bounded linear operators (Formula presented.) Motivated by a recent article, we obtain sufficient conditions so that for a compact operator (Formula presented.) and a closed subspace (Formula presented.) the following equation holds, which relates global approximation with local approximation: (Formula presented.) 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: (Formula presented.) for some (Formula presented.) satisfying (Formula presented.) where (Formula presented.) is the annihilator of Z. One such situation is when Z is an (Formula presented.) predual space and an (Formula presented.) ideal in Y and T is a multi-smooth operator of finite order. Another such situation is when X is an abstract (Formula presented.) 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. © 2022 Taylor & Francis Group, LLC.

Item Type: Journal Article
Publication: Numerical Functional Analysis and Optimization
Publisher: Taylor and Francis Ltd.
Additional Information: The copyright for this article belongs to the Author(s).
Keywords: Best approximation; Birkhoff–James orthogonality; distance formulae; linear operators; predual
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 31 Jan 2023 06:42
Last Modified: 31 Jan 2023 06:42
URI: https://eprints.iisc.ac.in/id/eprint/79601

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