Averaging theorems for linear operators in compact groups and semigroups

G. Murphy; T. West

Studia Mathematica (1997)

  • Volume: 124, Issue: 3, page 249-258
  • ISSN: 0039-3223

Abstract

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The Weyl criterion for uniform distribution of a sequence has an especially simple form in compact abelian groups. The authors use this and the structure of compact monothetic groups and semigroups to generalise the convergence, under certain compactness conditions, of the operator averages: n - 1 ∑ k = 1 n T k → P ( n → ∞ ) where P is a projection associated with the eigenvalue one of T.

How to cite

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Murphy, G., and West, T.. "Averaging theorems for linear operators in compact groups and semigroups." Studia Mathematica 124.3 (1997): 249-258. <http://eudml.org/doc/216412>.

@article{Murphy1997,
abstract = {The Weyl criterion for uniform distribution of a sequence has an especially simple form in compact abelian groups. The authors use this and the structure of compact monothetic groups and semigroups to generalise the convergence, under certain compactness conditions, of the operator averages: $n^\{-1\} ∑_\{k=1\}^n T^k → P (n→ ∞)$ where P is a projection associated with the eigenvalue one of T.},
author = {Murphy, G., West, T.},
journal = {Studia Mathematica},
keywords = {Weyl criterion; uniform distribution of a sequence; compact abelian groups; compact monothetic groups and semigroups; operator averages},
language = {eng},
number = {3},
pages = {249-258},
title = {Averaging theorems for linear operators in compact groups and semigroups},
url = {http://eudml.org/doc/216412},
volume = {124},
year = {1997},
}

TY - JOUR
AU - Murphy, G.
AU - West, T.
TI - Averaging theorems for linear operators in compact groups and semigroups
JO - Studia Mathematica
PY - 1997
VL - 124
IS - 3
SP - 249
EP - 258
AB - The Weyl criterion for uniform distribution of a sequence has an especially simple form in compact abelian groups. The authors use this and the structure of compact monothetic groups and semigroups to generalise the convergence, under certain compactness conditions, of the operator averages: $n^{-1} ∑_{k=1}^n T^k → P (n→ ∞)$ where P is a projection associated with the eigenvalue one of T.
LA - eng
KW - Weyl criterion; uniform distribution of a sequence; compact abelian groups; compact monothetic groups and semigroups; operator averages
UR - http://eudml.org/doc/216412
ER -

References

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  1. [1] G. Brown and W. Moran, Idempotents of compact monothetic semigroups, Proc. London Math. Soc. 22 (1971), 203-216. Zbl0215.49002
  2. [2] G. Brown and W. Moran, An unusual compact monothetic semigroup, Bull. London Math. Soc. 3 (1971), 291-296. Zbl0237.22003
  3. [3] B. Eckmann, Über monothetische Gruppen, Comment. Math. Helv. 16 (1944), 249-263. Zbl0061.04402
  4. [4] T. A. Gillespie and T. T. West, Operators generating weakly compact groups, Indiana Univ. Math. J. 21 (1972), 671-688. Zbl0233.47028
  5. [5] E. Hlawka, The Theory of Uniform Distribution, A B Academic Publishers, Berkhamsted, Herts, 1984. Zbl0563.10001
  6. [6] M. A. Kaashoek and T. T. West, Locally compact monothetic semi-algebras, Proc. London Math. Soc. 18 (1968), 428-438. Zbl0162.18601
  7. [7] K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63-97. Zbl0104.05501
  8. [8] W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1962. 
  9. [9] T. T. West, Weakly compact monothetic semigroups of operators in Banach spaces, Proc. Roy. Irish Acad. Sect. A 67 (1968), 27-37. Zbl0169.16701

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