2018/02/21 by Debmalya Sain, Sain, Debmalya
Computer Science · Mathematics · #46B04 #46B20 #47L05 #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1802.07527
openalex publication_date 2018/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the notion of approximate norm attainment set of a bounded\nlinear operator between Banach spaces and use it to obtain a complete\ncharacterization of smooth points in the space of compact linear operators,\nprovided the domain space is reflexive and Kadets-Klee. We also apply the\nconcept to characterize strong BPB property (sBPBp) of a pair of Banach spaces.\nWe further introduce uniform \ε- BPB approximation of a bounded linear\noperator and uniform strong BPB property (uniform sBPBp) with respect to a\ngiven family of norm one linear operators and explore some of the relevant\nproperties to illustrate its connection with earlier studies on\nBishop-Phelps-Bollob acuteas type theorems in Banach spaces. It is evident\nthat our study has deep connections with the study of smooth points in operator\nspaces. We obtain a complete characterization of uniform sBPBp for a pair of\nBanach spaces, with respect to a given family of norm one bounded linear\noperators between them. As the final result of this paper, we prove that if \n mathbbX is a reflexive Kadets-Klee Banach space and mathbbY is any\nBanach space, then the pair ( mathbbX, mathbbY) has sBPBp for compact\noperators. Our results extend, complement and improve some of the earlier\nresults in this context.\n