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arXiv · 1806.10876

A complete characterization of smoothness in the space of bounded linear operators

Debmalya Sain·Kallol Paul·Arpita Mal·Anubhab Ray
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Abstract

We completely characterize smoothness of bounded linear operators between infinite dimensional real normed linear spaces, probably for the very first time, by applying the concepts of Birkhoff-James orthogonality and semi-inner-products in normed linear spaces. In this context, the key aspect of our study is to consider norming sequences for a bounded linear operator, instead of norm attaining elements. We also obtain a complete characterization of smoothness of bounded linear operators between real normed linear spaces, when the corresponding norm attainment set non-empty. This illustrates the importance of the norm attainment set in the study of smoothness of bounded linear operators. Finally, we prove that G\^{a}teaux differentiability and Fr\'{e}chet differentiability are equivalent for compact operators in the space of bounded linear operators between a reflexive Kadets-Klee Banach space and a Fr\'{e}chet differentiable normed linear space, endowed with the usual operator norm.

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Debmalya Sain, Kallol Paul, Arpita Mal, Anubhab Ray. 2018-06-28. A complete characterization of smoothness in the space of bounded linear operators. https://doi.org/10.1080/03081087.2019.1586824

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