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# Journal of Operator Theory

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

Polynomial conditions on operator semigroups

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 $f(xy, yx)$.