Minimal reducing subspaces of the unilateral shift operator on an operator weighted sequence space (Q1881029)

Let \(K\) be a separable complex Hilbert space and let \(\{ B_{n}\}_{n=0}^{\infty }\) be a sequence of positive invertible selfadjoint operators on \(K\). Suppose that this sequence and the sequence \(\{ B_{n}^{-1}\}_{n=0}^{\infty }\) are bounded. Denote by \(H^{2}(B)\) the Hilbert space of all sequences \(f=\{ f_{n} \} \) (\(f_{n}\in K\) for all \(n\)) with finite norm \( \| f \|^{2} = \sum_{n=0}^{\infty }\| f_{n} \|^{2}. \) Denote by \(S\) the unilateral shift on \(H^{2}(B)\). The present paper is primarily concerned with specifying the minimal reducing subspaces of \(S\). This is achieved under the additional assumption that all \(B_{n}\)'s are diagonal in some fixed orthonormal basis in \(K\) (one ought to note that in this case \(H^{2}(B)\) is the direct sum of an infinite family of (scalar) weighted \(l^{2}\)-spaces). The main results are formulated in terms of diagonal elements of matrices representing weight operators.