Strongly continuous semigroups of operators generated by systems of pseudodifferential operators in weighted L p -spaces

Abstract

This paper considers semigroups of operators generated by pseudodifferential operators in weighted L p -spaces of vector functions on \( {\mathbb{R}^n} \) (or on a compact manifold without boundary). Sufficient conditions for a semigroup to be strongly continuous and analytic are obtained, conditions for it to be completely continuous are found, and the distribution of the eigenvalues of its infinitesimal generator is examined. Also, an integral representation that singles out the principal term of the semigroup as t → 0+ is established.