Characterizations of Sunduals (Summary)

Characterizations of Sunduals (Summary) Positivity 7: 81–85, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands. W.A.J. LUXEMBURG Department of Mathematics, 253–37, California Institute of Technology, Pasadena, CA 91125-003, USA. E-mail: lux@caltech.edu 1. Let G ={T(t) : t  0} be a C -semigroup of bounded linear operators, acting on a real or complex Banach space E, i.e., T(0) = I , the identity operator; (∀t, s 0)(T (t +s) = T(t)T (s)); and the mapping t → T(t) of R := {t ∈ R : t  0} into L(E) the Banach algebra of all bounded linear operators acting on E is strongly continuous (in this context “C " means continuous at 0). ∗ ∗ The dual semigroup G := {T (t ) : t  0}, deﬁnes a semigroup of bounded linear operators on the Banach dual space E of E.If E is not reﬂexive, then G need not be strongly continuous. By its very deﬁnition, however, the mapping ∗ ∗ t → T (t ) of R into L(E ) is a weak-∗-continuous semigroup of bounded and weak-∗-continuous linear operators of E . For example, let E = C (R), the space of the continuous functions on R vanishing http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Characterizations of Sunduals (Summary)

, Volume 7 (2) – Oct 17, 2004
5 pages

/lp/springer_journal/characterizations-of-sunduals-summary-fZgF5zb0XT
Publisher
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1025820100268
Publisher site
See Article on Publisher Site

Abstract

Positivity 7: 81–85, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands. W.A.J. LUXEMBURG Department of Mathematics, 253–37, California Institute of Technology, Pasadena, CA 91125-003, USA. E-mail: lux@caltech.edu 1. Let G ={T(t) : t  0} be a C -semigroup of bounded linear operators, acting on a real or complex Banach space E, i.e., T(0) = I , the identity operator; (∀t, s 0)(T (t +s) = T(t)T (s)); and the mapping t → T(t) of R := {t ∈ R : t  0} into L(E) the Banach algebra of all bounded linear operators acting on E is strongly continuous (in this context “C " means continuous at 0). ∗ ∗ The dual semigroup G := {T (t ) : t  0}, deﬁnes a semigroup of bounded linear operators on the Banach dual space E of E.If E is not reﬂexive, then G need not be strongly continuous. By its very deﬁnition, however, the mapping ∗ ∗ t → T (t ) of R into L(E ) is a weak-∗-continuous semigroup of bounded and weak-∗-continuous linear operators of E . For example, let E = C (R), the space of the continuous functions on R vanishing

Journal

PositivitySpringer Journals

Published: Oct 17, 2004

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