Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators

Pseudodifferential Operators and Regularized Traces

This is a survey on trace constructions on various operator algebras with an emphasis on regularized traces on algebras of pseudodifferential operators. For motivation our point of departure is the classical Hilbert space trace which is the unique semifinite normal trace on the algebra of bounded operators on a separable Hilbert space. Dropping the normality assumption leads to the celebrated D...

Perturbation Theorems for Convoluted C-semigroups and Cosine Functions

On the Compactness of Strongly Continuous Semigroups and Cosine Functions of Operators

In this note we relate two notions of compactness for strongly continuous semigroups of linear operators and cosine functions of linear operators. Specifically, if T denotes a strongly continuous semigroup of linear operators defined on a Banach space X, we will show that T is compact if and only if the set {(T(-)x : x £ X, \\x\\ < 1} is relatively compact in any space LP([Q, a]) ; X) for 1 < p...

Pseudodifferential Operators

The study of pseudodifferential operators emerged in the 1960’s, having its origins in the study of singular integro-differential operators. In fact, Friedrichs and Lax coined the term “pseudodifferential operator” in their 1965 paper entitled “Boundary Value Problems for First Order Operators”. Since that time, pseudodifferential operators have proved useful in many arenas of modern analysis a...

Convoluted C-cosine Functions and Convoluted C-semigroups