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2000 | 139 | 1 | 91-100
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On operator bands

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A multiplicative semigroup of idempotent operators is called an operator band. We prove that for each K>1 there exists an irreducible operator band on the Hilbert space $l^2$ which is norm-bounded by K. This implies that there exists an irreducible operator band on a Banach space such that each member has operator norm equal to 1. Given a positive integer r, we introduce a notion of weak r-transitivity of a set of bounded operators on a Banach space. We construct an operator band on $l^2$ that is weakly r-transitive and is not weakly (r+1)-transitive. We also study operator bands S satisfying a polynomial identity p(A, B) = 0 for all non-zero A,B ∈ S, where p is a given polynomial in two non-commuting variables. It turns out that the polynomial $p(A, B) = (A B - B A)^2$ has a special role in these considerations.
Czasopismo
Rocznik
Tom
139
Numer
1
Strony
91-100
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-05-24
poprawiono
2000-01-04
Twórcy
autor
  • Department of Math and CS, University of Prince Edward Island, Charlottetown, PEI C1A 4P3, Canada. , gmacdonald@upei.ca
autor
  • Department of Math, Stats and CS, Dalhousie University, Halifax, NS B3H 3J3, Canada. , radjavi@mscs.dal.ca
autor
Bibliografia
  • [1] J. B. Conway, A Course in Functional Analysis, Springer, 1990.
  • [2] R. Drnovšek, An irreducible semigroup of idempotents, Studia Math. 125 (1997), 97-99.
  • [3] P. Fillmore, G. W. MacDonald, M. Radjabalipour and H. Radjavi, Principal-ideal bands, Semigroup Forum 59 (1999), 362-373.
  • [4] J. A. Green and D. Rees, On semigroups in which $x^r = x$, Proc. Cambridge Philos. Soc. 48 (1952), 35-40.
  • [5] L. Livshits, G. W. MacDonald, B. Mathes and H. Radjavi, Reducible semigroups of idempotent operators, J. Operator Theory 40 (1998), 35-69.
  • [6] L. Livshits, G. W. MacDonald, B. Mathes and H. Radjavi, On band algebras, ibid., to appear.
  • [7] M. Petrich, Lectures in Semigroups, Akademie-Verlag, Berlin, and Wiley, London, 1977.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv139i1p91bwm
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