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# Journal of Operator Theory

Volume 64, Issue 2, Fall 2010  pp. 245-264.

Nil polynomials and reducibility of operator semigroups

Summary:  If $\mathcal{S}$ is a multiplicative semigroup of bounded operators on a Banach space, what is the effect of a polynomial identity on reducibility of $\mathcal{S}$, i.e., the existence of a closed invariant subspace for $\mathcal{S}$? More specifically, which noncommutative polynomials in two variables have the property that whenever $f(A,B)=0$, or more generally, $% f(A,B)$ is quasinilpotent for all $A$ and $B$ in $\mathcal{S}$, then $\mathcal{S}$ is reducible or possibly (simultaneously) triangularizable? A well-known example of such polynomials that works at least for semigroups of compact operators is $f(x,y)=xy-yx$. Extensions of this result are also known for certain classes of polynomials that yield reducibility and triangularizability. We study this question for arbitrary homogeneous polynomials and present fairly general reducing and triangularizing conditions. As a corollary, we obtain polynomial conditions under which every compact group is necessarily Abelian.