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On operator bands

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Drnovšek R., Livshits L., MacDonald G. W., Mathes B., Radjavi H., Šemrl P.

Studia Mathematica

2000

tom 139

nr 1

91-100

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.

Ograniczanie wyników

1 Studia Mathematica

1 Drnovšek R.

1 Livshits L.

1 MacDonald G. W.

1 Mathes B.

1 Radjavi H.

1 Šemrl P.

1 2000

Wyniki wyszukiwania

On operator bands