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SMOOTH POINTS IN OPERATOR SPACES AND SOME BISHOP-PHELPS-BOLLOBAS TYPE THEOREMS IN BANACH SPACES

Sain, Debmalya (2019) SMOOTH POINTS IN OPERATOR SPACES AND SOME BISHOP-PHELPS-BOLLOBAS TYPE THEOREMS IN BANACH SPACES. In: OPERATORS AND MATRICES, 13 (2). pp. 433-445.

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Official URL: https://dx.doi.org/10.7153/oam-2019-13-32

Abstract

We introduce the notion of approximate norm attainment set of a bounded linear operator between Banach spaces and use it to obtain a complete characterization of smooth points in the space of compact linear operators, provided the domain space is reflexive and Kadets-Klee. We also apply the concept to characterize strong BPB property (sBPBp) of a pair of Banach spaces. We further introduce uniform epsilon-BPB approximation of a bounded linear operator and uniform strong BPB property (uniform sBPBp) with respect to a given family of norm one linear operators and explore some of the relevant properties to illustrate its connection with earlier studies on Bishop-Phelps-Bollobas type theorems in Banach spaces. It is evident that our study has deep connections with the study of smooth points in operator spaces. We obtain a complete characterization of uniform sBPBp for a pair of Banach spaces, with respect to a given family of norm one bounded linear operators between them. As the final result of this paper, we prove that if X is a reflexive Kadets-Klee Banach space and Y is any Banach space, then the pair (X, Y) has sBPBp for compact operators. Our results extend, complement and improve some of the earlier results in this context.

Item Type: Journal Article
Additional Information: copyright for this article belongs to ELEMENT
Keywords: Banach space; norm attainment; Bishop-Phelps-Bollobas property; smooth operators
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Depositing User: Id for Latest eprints
Date Deposited: 19 Aug 2019 10:35
Last Modified: 19 Aug 2019 10:35
URI: http://eprints.iisc.ac.in/id/eprint/63470

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