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

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.