Semigroups of Linear Operators and Applications to Partial Differential Equations pp 42-75 | Cite as

# Spectral Properties and Regularity

Chapter

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## Abstract

Let where

*T*(*t*) be a C_{0}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)

*w*— lim denotes the weak limit in*X*. The domain of*Ã*is the set of all x ϵ*X*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 Ã 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.## Keywords

Banach Space Linear Operator Spectral Property Bounded Linear Operator Weak Limit
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag New York, Inc. 1983