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

Abstract

Let $$(S, {\\mathcal {B}}, m)$$\n be a finite measure space. In this paper we show that every bounded linear operator T from $$L^{p_{1}}(S)$$\n into $$L^{p_{2}}(S)$$\n is an S-operator (or a generalized pseudo-differential operator) with the symbol $$\\sigma $$\n for some $$ 1\\le \\alpha<p_{1}, p_{2}<\\beta \\le \\infty $$\n . 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 $$L^p(S)$$\n 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 $$L^{2}(S)$$\n are compact, self-adjoint, or compact self-adjoint. Lastly, we give a result concerning to factorization of compact operators.