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Self-adjointness and Compactness of Operators Related to Finite Measure Spaces

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Abstract

Let \((S, {\mathcal {B}}, m)\) be a finite measure space. In this paper we show that every bounded linear operator T from \(L^{p_{1}}(S)\) into \(L^{p_{2}}(S)\) is an S-operator (or a generalized pseudo-differential operator) with the symbol \(\sigma \) for some \( 1\le \alpha<p_{1}, p_{2}<\beta \le \infty \). We make use of this symbolic representation to study various functional analytical properties of T. First, we present necessary and sufficient conditions for a function to be the symbol of the adjoint of T in terms of the symbol of T. Then, we give necessary and sufficient conditions to guarantee that the bounded linear operators on \(L^p(S)\) posses a particular complex number (function) as its eigenvalue (eigenfunction). As an application, we obtain necessary and sufficient conditions on the symbols to ensure that corresponding bounded linear operators on \(L^{2}(S)\) are compact, self-adjoint, or compact self-adjoint. Lastly, we give a result concerning to factorization of compact operators.

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Acknowledgements

The authors would like to thank the anonymous referees for his/her valuable suggestions. Shyam Swarup Mondal gratefully acknowledges the support provided by IIT Guwahati, Government of India. He also thanks his supervisor Jitendriya Swain for his encouragement. Vishvesh Kumar is supported by FWO Odysseus 1 Grant G.0H94.18N: Analysis and Partial Differential Equations.

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Authors and Affiliations

  1. Department of Mathematics, IIT Guwahati, Guwahati, Assam, India

    Shyam Swarup Mondal

  2. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000, Ghent, Belgium

    Vishvesh Kumar

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  1. Shyam Swarup Mondal
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Correspondence to Shyam Swarup Mondal.

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Communicated by H. Turgay Kaptanoglu.

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This article is part of the topical collection “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Aurelian Gheondea and Serap Oztop.

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Mondal, S.S., Kumar, V. Self-adjointness and Compactness of Operators Related to Finite Measure Spaces. Complex Anal. Oper. Theory 15, 22 (2021). https://doi.org/10.1007/s11785-020-01067-2

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