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Spectral Properties and Regularity

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

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.

Keywords

  • Banach Space
  • Linear Operator
  • Spectral Property
  • Bounded Linear Operator
  • Weak Limit

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© 1983 Springer-Verlag New York, Inc.

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Pazy, A. (1983). Spectral Properties and Regularity. In: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol 44. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5561-1_2

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  • DOI: https://doi.org/10.1007/978-1-4612-5561-1_2

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-5563-5

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