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Some Bishop–Phelps–Bollobás type properties in Banach spaces with respect to minimum norm of bounded linear operators

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Abstract

In this paper, we study a Bishop–Phelps–Bollobás type property called the property \({\mathbf{L}}_{o,o}\) of a pair of Banach spaces. Getting motivated by this, we introduce the notion of Approximate minimizing property (AMp) of a pair of Banach spaces and characterize finite dimensionality of Banach spaces with respect to this property. We further introduce the notion of approximate minimum norm attainment set of a bounded linear operator and characterize the AMp with the help of Hausdorff convergence of the sequence of approximate minimum norm attainment sets of bounded linear operators. We also investigate sufficient conditions for the holding of some weaker forms of the AMp for a pair of Banach spaces. Finally, we define and study uniform \(\varepsilon\)-approximation of a bounded linear operator in terms of its minimum norm.

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Acknowledgements

The author expresses his thankfulness to both the anonymous referees for their valuable suggestions to improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Assam University, Silchar, Assam, 788011, India

    Uday Shankar Chakraborty

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  1. Uday Shankar Chakraborty
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Correspondence to Uday Shankar Chakraborty.

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Communicated by Jacek Chmielinski.

To my parents.

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Chakraborty, U.S. Some Bishop–Phelps–Bollobás type properties in Banach spaces with respect to minimum norm of bounded linear operators. Ann. Funct. Anal. 12, 46 (2021). https://doi.org/10.1007/s43034-021-00132-x

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