Spectral Properties and Regularity

Abstract

Let T(t) be a C₀ semigroup of bounded linear operators on a Banach space X. Let A be its infinitesimal generator as defined in Definition 1.1.1. We consider now the operator

$$\tilde{A}x = w - \mathop{{\lim }}\limits_{{h \downarrow 0}} \frac{{T(h)x - x}}{h}$$

(1.1)

where w — lim denotes the weak limit in X. The domain of à is the set of all x ϵX forr which the weak limit on the right-hand side of (1.1) exists. Since the existence of a limit implies the existence of a weak limit, it is clear that à extends A. That this extension is not genuine follows from Theorem 1.3 below. In the proof of this theorem we will need the following real variable results.

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