Spectral Properties and Regularity

Part of the Applied Mathematical Sciences book series (AMS, volume 44)


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


Banach Space Linear Operator Spectral Property Bounded Linear Operator Weak Limit 
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Copyright information

© Springer-Verlag New York, Inc. 1983

Authors and Affiliations

  • A. Pazy
    • 1
  1. 1.Planning and Budgeting CommitteeCouncil for Higher EducationJerusalemIsrael

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