The Functional Calculus on Hilbert Spaces

Abstract

We start with providing necessary background information on the functional calculus on Hilbert spaces. In Section 7.1 we show how numerical range conditions account for the boundedness of the H∞-calculus. In the sector case this is essentially von Neumann’s inequality (Section 7.1.3), in the strip case it is a result by Crouzeix and Delyon (Section 7.1.5). Prom von Neumann’s inequality we obtain certain ‘mapping theorems for the numerical range’ (Section 7.1.4). Section 7.2 is devoted to C0-groups on Hilbert spaces. We discuss Liapunov’s direct method (Section 7.2.1) from Linear Systems Theory and apply it to obtain a remarkable decomposition and similarity result for group generators (Section 7.2.2). This allows us to prove a theorem of de Laubenfels and Boyadzhiev on the boundedness of the H∞-calculus on strips for such operators. This result can be approached also in a different way, yielding in addition a characterisation of group generators (Section 7.2.3). Section 7.3 is devoted mainly to the connection of the functional calculus with similarity theorems. The main tool is McIntosh’s fundamental result on the boundedness of the H∞-calculus (Section 7.3.1). The similarity questions we are interested in deal with operators defined by sesquilinear forms. We introduce these operators in Section 7.3.2 and obtain in Section 7.3.3 a characterisation of such operators up to similarity. Afterwards, several theorems on similarity are proved, related also to the so-called square root problem. We give an example of a C0-semigroup which is not similar even to a quasi-contractive semigroup (Section 7.3.4). Finally, we present applications to generators of cosine functions, showing in particular that after a similarity transformation those operators always have numerical range in a horizontal parabola (Section 7.4).