On operator bands
Drnovšek, Roman ; Livshits, Leo ; MacDonald, Gordon ; Mathes, Ben ; Radjavi, Heydar ; Šemrl, Peter
Studia Mathematica, Tome 141 (2000), p. 91-100 / Harvested from The Polish Digital Mathematics Library
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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 l2 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 l2 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)=(AB-BA)2 has a special role in these considerations.

Publié le : 2000-01-01
EUDML-ID : urn:eudml:doc:216713 
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Drnovšek, Roman; Livshits, Leo; MacDonald, Gordon; Mathes, Ben; Radjavi, Heydar; Šemrl, Peter. On operator bands. Studia Mathematica, Tome 141 (2000) pp. 91-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv139i1p91bwm/

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