Strong continuity of semigroup homomorphisms

Bolis Basit; A. Pryde

Studia Mathematica (1999)

  • Volume: 132, Issue: 1, page 71-78
  • ISSN: 0039-3223

Abstract

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Let J be an abelian topological semigroup and C a subset of a Banach space X. Let L(X) be the space of bounded linear operators on X and Lip(C) the space of Lipschitz functions ⨍: C → C. We exhibit a large class of semigroups J for which every weakly continuous semigroup homomorphism T: J → L(X) is necessarily strongly continuous. Similar results are obtained for weakly continuous homomorphisms T: J → Lip(C) and for strongly measurable homomorphisms T: J → L(X).

How to cite

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Basit, Bolis, and Pryde, A.. "Strong continuity of semigroup homomorphisms." Studia Mathematica 132.1 (1999): 71-78. <http://eudml.org/doc/216586>.

@article{Basit1999,
abstract = {Let J be an abelian topological semigroup and C a subset of a Banach space X. Let L(X) be the space of bounded linear operators on X and Lip(C) the space of Lipschitz functions ⨍: C → C. We exhibit a large class of semigroups J for which every weakly continuous semigroup homomorphism T: J → L(X) is necessarily strongly continuous. Similar results are obtained for weakly continuous homomorphisms T: J → Lip(C) and for strongly measurable homomorphisms T: J → L(X).},
author = {Basit, Bolis, Pryde, A.},
journal = {Studia Mathematica},
keywords = {representation; semigroup homomorphism; weak continuity; strong continuity; Lipschitz map; abelian topological semigroups; Lipschitz functions},
language = {eng},
number = {1},
pages = {71-78},
title = {Strong continuity of semigroup homomorphisms},
url = {http://eudml.org/doc/216586},
volume = {132},
year = {1999},
}

TY - JOUR
AU - Basit, Bolis
AU - Pryde, A.
TI - Strong continuity of semigroup homomorphisms
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 1
SP - 71
EP - 78
AB - Let J be an abelian topological semigroup and C a subset of a Banach space X. Let L(X) be the space of bounded linear operators on X and Lip(C) the space of Lipschitz functions ⨍: C → C. We exhibit a large class of semigroups J for which every weakly continuous semigroup homomorphism T: J → L(X) is necessarily strongly continuous. Similar results are obtained for weakly continuous homomorphisms T: J → Lip(C) and for strongly measurable homomorphisms T: J → L(X).
LA - eng
KW - representation; semigroup homomorphism; weak continuity; strong continuity; Lipschitz map; abelian topological semigroups; Lipschitz functions
UR - http://eudml.org/doc/216586
ER -

References

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  1. [1] B. Basit and A. J. Pryde, Differences of vector-valued functions on topological groups, Proc. Amer. Math. Soc. 124 (1996), 1969-1975. Zbl0852.43001
  2. [2] B. Basit and A. J. Pryde, Unitary eigenvalues of semigroup homomorphisms, Monash University Analysis Paper 105, May 1997, 8 pp. 
  3. [3] J. P. R. Christensen, Joint continuity of separately continuous functions, Proc. Amer. Math. Soc. 82 (1981), 455-461. Zbl0472.54007
  4. [4] C. Datry et G. Muraz, Analyse harmonique dans les modules de Banach I: propriétés générales, Bull. Sci. Math. 119 (1995), 299-337. Zbl0840.43005
  5. [5] N. Dunford, On one parameter groups of linear transformations, Ann. of Math. 39 (1938), 569-573. Zbl0019.26604
  6. [6] J. A. Goldstein, Extremal properties of contraction semigroups on Hilbert and Banach spaces, Bull. London Math. Soc. 25 (1993), 369-376. Zbl0794.47025
  7. [7] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., 1957. Zbl0078.10004
  8. [8] I. Namioka, Separate continuity and joint continuity, Pacific J. Math. 51 (1974), 515-531. Zbl0294.54010
  9. [9] W. Rudin, Fourier Analysis on Groups, Interscience, 1967. 
  10. [10] K. Yosida, Functional Analysis, Springer, 1966. 

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