Minimal reducing subspaces of an operator-weighted shift

Abstract

We introduce a family $\mathcal{T}$ consisting of invertible matrices with exactly one nonzero entry in each row and each column. The elements of $\mathcal{T}$ are possibly mutually noncommuting, and they need not be normal or self-adjoint. We consider an operator-valued unilateral weighted shift $W$ with a uniformly bounded sequence of weights belonging to $\mathcal{T}$, and we describe its minimal reducing subspaces.