Journal of Operator Theory

Volume 53, Issue 1, Winter 2005 pp. 197-220.

Polynomial conditions on operator semigroups

Authors: Heydar Radjavi
Author institution: Department of Pure Mathematics, University of Waterloo, Waterloo, N2L 3G1, Canada

Summary: It is known that if

$AB -BA$ is quasinilpotent for every

$A$ and

$B$ in a multiplicative semigroup

${\cS}$ of compact operators on a complex Banach space, then

${\cS}$ is triangularizable. Possible extensions of this result are examined when

$AB-BA$ is replaced with a general noncommutative polynomial in

$A$ and

$B$ . Easily checkable conditions on polynomials are found which enable us to reduce the problem to the case of finite groups acting on finite-dimensional spaces. In particular, all homogeneous noncommutative polynomials

$f$ in two variables with the following property are determined: if

$f(A, B)$ is quasinilpotent for all

$A$ and

$B$ in

${\cS}$ , then

${\cS}$ has a chain of invariant subspaces such that every induced semigroup on a ``gap'' of the chain is a matrix group that is finite modulo its centre. A triangularizability theorem which is a direct generalization of the known result on

$AB -BA$ mentioned above, is obtained by replacing the polynomial

$xy -yx$ with suitable polynomials of the form

$f(xy, yx)$ .

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