Generalized Spectrality and Interpolation of Germs of Analytic Functions

Abstract

In this Lecture we describe those cases when a “spectral decomposition” of the model operator T_Θ is possible, taking these words in a broader sense than in Lectures VI–VIII. In place of eigenspaces we will consider arbitrary spectral (and even more general) subspaces and the corresponding parts of the spectrum σ(T_Θ) will be drawn from a specially chosen σ-algebra of Borel sets. We must of course maintain sufficiently tight connections with the classical expansion as provided by the Spectral Theorem for normal operators. The necessary language will be borrowed from the theory of spectral operators in the sense of Dunford, concerning which some notions will be developed in the “Concluding Notes” of this Lecture.