On polynomial roots of class A operators

In this paper, we show that if T ∈ 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.