On operator bands
Description
Citation
| Title | On operator bands |
| Author(s) | R. Drnovsek, L. Livshits, G. MacDonald, B. Mathes, H. Radjavi, P. Semrl |
| Journal | Studia Mathematica |
| Date | 2000 |
| Volume | 139 |
| Issue | 1 |
| Start page | 91 |
| End page | 100 |
| Abstract | 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 epsilon 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. |
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