Pseudo-differential analysis of bounded linear operators from $$L^{p_1}({\mathbb S}^1)$$ into $$L^{p_2}({\mathbb S}^1)$$

Abstract

In this paper, we first show that every bounded linear operator T from $$ L^{p_{1}}({\mathbb S}^1) $$ into $$ L^{p_{2}}({\mathbb S}^1) $$ for $$1< p_{1}, p_{2}<\infty $$ is a pseudo-differential operator and give a formula for its symbol $$ \sigma $$ . Using the pseudo-differential representation of T, we offer necessary and sufficient conditions on the symbols $$ \sigma $$ to find the symbol of the adjoint $$ T^{*}$$ . We present necessary and sufficient conditions on the symbols $$ \sigma $$ to ensure that their corresponding bounded linear operators from $$ L^{2}({\mathbb S}^1) $$ into $$ L^{2}({\mathbb S}^1)$$ are self-adjoint, compact or compact self-adjoint. As applications, in case $$ \sigma $$ is a real-valued function, we show that the corresponding bounded linear operator T is self-adjoint if and only if $$ \sigma $$ depends only on one variable. Also, we prove that every compact operator from $$ L^{2}({\mathbb S}^1) $$ into $$ L^{2}({\mathbb S}^1) $$ is the product of two compact operators and give some formulas for the symbols of compact and compact self-adjoint operators.