On polynomial roots of class A operators

Abstract

In this paper, we show that if $T\in{\cal L(H)}$ is a polynomial root of a class $A$ operator, i.e. $p(T)$ belongs to class $A$ for some nonconstant polynomial $p$, then $T$ is subscalar. As a corollary, we obtain that such an operator with thick spectrum has a nontrivial invariant subspace. We also prove that an algebraic extension of a polynomial root of a class $A$ operator is subscalar
and provide its several properties.