Spectral Families of Projections, Semigroups, and Differential Operators

Abstract

This paper presents new developments in abstract spectral theory suitable for treating classical differential and translation operators. The methods are specifically geared to conditional convergence such as arises in Fourier expansions and in Fourier inversion in general. The underlying notions are spectral family of projections and well-bounded operator, due to D. R. Smart and J. R. Ringrose. The theory of well-bounded operators is considerably expanded by the introduction of a class of operators with a suitable polar decomposition. These operators, called polar operators, have a canonical polar decomposition, are free from restrictions on their spectra (in contrast to well-bounded operators), and lend themselves to semigroup considerations. In particular, a generalization to arbitrary Banach spaces of Stone’s theorem for unitary groups is obtained. The functional calculus for well-bounded operators with spectra in a nonclosed arc is used to study closed, densely defined operators with a well-bounded resolvent. Such an operator $L$ is represented as an integral with respect to the spectral family of its resolvent, and a sufficient condition is given for $(- L)$ to generate a strongly continuous semigroup. This approach is applied to a large class of ordinary differential operators. It is shown that this class contains significant subclasses of operators which have a polar resolvent or generate strongly continuous semigroups. Some of the latter consist of polar operators up to perturbation by a semigroup continuous in the uniform operator topology.