Geodesic
On operator bands
Studia Mathematica, Tome 139 (2000) no. 1, pp. 91-100

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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.
DOI : 10.4064/sm-139-1-91-100
Keywords: invariant subspaces, idempotents, operator semigroups

Roman Drnovšek 1 ; Leo Livshits 1 ; Gordon W. MacDonald 1 ; Ben Mathes 1 ; Heydar Radjavi 1 ; Peter Šemrl 1

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Roman Drnovšek; Leo Livshits; Gordon W. MacDonald; Ben Mathes; Heydar Radjavi; Peter Šemrl. On operator bands. Studia Mathematica, Tome 139 (2000) no. 1, pp. 91-100. doi: 10.4064/sm-139-1-91-100

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