The Boundedness of the H^∞-Calculus

Abstract

This chapter mainly provides technical background information. We start with the so-called Convergence Lemma (Section 5.1) and some fundamental boundedness and approximation results (Section 5.2). Then we prove equivalence of boundedness of F-functional calculi for different subalgebras F of H^∞ (Section 5.3). We also introduce and study the H^∞-angle (Section 5.4) and present permanence results with respect to additive perturbations (Section 5.5). Finally, we prove a technical lemma which indicates the connections with Harmonic Analysis (Section 5.6).