On operator bands

• Autorzy: Roman Drnovšek , Leo Livshits , Gordon W. MacDonald , Ben Mathes , Heydar Radjavi , Peter Šemrl
• 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.

Słowa kluczowe

EN

invariant subspaces   idempotents   operator semigroups  

Kategorie tematyczne

• 47D03: Groups and semigroups of linear operators
• 47A15: Invariant subspaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2000
• Tom: 139
• Numer: 1
• Strony: 91-100

Daty

wydano

2000

otrzymano

1999-05-24

poprawiono

2000-01-04

Twórcy

autor

Roman Drnovšek

• Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia ,

autor

Leo Livshits

• Department of Math and CS, Colby College Waterville, ME 04901, U.S.A. ,

autor

Gordon W. MacDonald

• Department of Math and CS, University of Prince Edward Island, Charlottetown, PEI C1A 4P3, Canada. ,

autor

Ben Mathes

• Department of Math and CS, Colby College Waterville, ME 04901, U.S.A. ,

autor

Heydar Radjavi

• Department of Math, Stats and CS, Dalhousie University, Halifax, NS B3H 3J3, Canada. ,

autor

Peter Šemrl

• Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia ,

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.