On operator bands

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.PT: J; CR: CONWAY JB, 1990, COURSE FUNCTIONAL AN DRNOVSEK R, 1997, STUD MATH, V125, P97 FILLMORE P, 1999, SEMIGROUP FORUM, V59, P362 GREEN JA, 1952, P CAMBRIDGE PHILOS S, V48, P35 LIVSHITS L, IN PRESS J OPERATOR LIVSHITS L, 1998, J OPERAT THEOR, V40, P35 PETRICH M, 1977, LECT SEMIGROUPS; NR: 7; TC: 0; J9: STUD MATH; PG: 10; GA: 348YKSource type: Electronic(1