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Self-adjointness and compactness of operators related to finite measure spaces

Shyam Swarup Mondal and Vishvesh Kumar (UGent)
(2021) COMPLEX ANALYSIS AND OPERATOR THEORY. 15(2).
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Abstract
Let (S, B, m) be a finite measure space. In this paper we show that every bounded linear operator T from L-p1(S) into L-p2(S) is an S-operator (or a generalized pseudo-differential operator) with the symbol sigma for some 1 <= alpha < p(1), p(2) < beta <= infinity. 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.
Keywords
Computational Theory and Mathematics, Applied Mathematics, Computational Mathematics, Pseudo-differential operators, Finite measure space, Fourier transform, S-operators, Self-adjoint operators, Compact operators, Eigenvalues, Eigenfunctions

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MLA
Mondal, Shyam Swarup, and Vishvesh Kumar. “Self-Adjointness and Compactness of Operators Related to Finite Measure Spaces.” COMPLEX ANALYSIS AND OPERATOR THEORY, vol. 15, no. 2, 2021, doi:10.1007/s11785-020-01067-2.
APA
Mondal, S. S., & Kumar, V. (2021). Self-adjointness and compactness of operators related to finite measure spaces. COMPLEX ANALYSIS AND OPERATOR THEORY, 15(2). https://doi.org/10.1007/s11785-020-01067-2
Chicago author-date
Mondal, Shyam Swarup, and Vishvesh Kumar. 2021. “Self-Adjointness and Compactness of Operators Related to Finite Measure Spaces.” COMPLEX ANALYSIS AND OPERATOR THEORY 15 (2). https://doi.org/10.1007/s11785-020-01067-2.
Chicago author-date (all authors)
Mondal, Shyam Swarup, and Vishvesh Kumar. 2021. “Self-Adjointness and Compactness of Operators Related to Finite Measure Spaces.” COMPLEX ANALYSIS AND OPERATOR THEORY 15 (2). doi:10.1007/s11785-020-01067-2.
Vancouver
1.
Mondal SS, Kumar V. Self-adjointness and compactness of operators related to finite measure spaces. COMPLEX ANALYSIS AND OPERATOR THEORY. 2021;15(2).
IEEE
[1]
S. S. Mondal and V. Kumar, “Self-adjointness and compactness of operators related to finite measure spaces,” COMPLEX ANALYSIS AND OPERATOR THEORY, vol. 15, no. 2, 2021.
@article{8713871,
  abstract     = {{Let (S, B, m) be a finite measure space. In this paper we show that every bounded linear operator T from L-p1(S) into L-p2(S) is an S-operator (or a generalized pseudo-differential operator) with the symbol sigma for some 1 <= alpha < p(1), p(2) < beta <= infinity. 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.}},
  articleno    = {{22}},
  author       = {{Mondal, Shyam Swarup and Kumar, Vishvesh}},
  issn         = {{1661-8254}},
  journal      = {{COMPLEX ANALYSIS AND OPERATOR THEORY}},
  keywords     = {{Computational Theory and Mathematics,Applied Mathematics,Computational Mathematics,Pseudo-differential operators,Finite measure space,Fourier transform,S-operators,Self-adjoint operators,Compact operators,Eigenvalues,Eigenfunctions}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{30}},
  title        = {{Self-adjointness and compactness of operators related to finite measure spaces}},
  url          = {{http://doi.org/10.1007/s11785-020-01067-2}},
  volume       = {{15}},
  year         = {{2021}},
}
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