On the topology of invariant subspaces of a shift of higher multiplicity (Q361906)

The author gives a description of continuous paths in the invariant subspace lattice of the unilateral shift operator of arbitrary finite multiplicity. The main result of the paper reads as follows. {Theorem.} Let \(H^{2}({\mathbb C}^{n})\) be the Hardy space for some finite \(n\) and let \(\{ p_{t}\}_{t\in[0,1]}\) be a continuous in the uniform operator topology family of orthogonal projections on \(H^{2}({\mathbb C}^{n})\) such that \(p_{t}H^{2}({\mathbb C}^{n})\), \(t\in[0,1]\), are invariant subspaces of the unilateral shift operator of multiplicity \(n\). Then there exists an integer \(m\leq n\) and a family of inner operator-valued functions \( G_{t}\in H^{\infty}({\mathbb T},B({\mathbb C}^{m},{\mathbb C}^{n}))\), \(t\in[0,1]\), such that \(p_{t}H^{2}({\mathbb C}^{n})=G_{t}H^{2}({\mathbb C}^{m})\) and \(\{ G_{t}\}_{t\in[0,1]} \) is \(\sup\)-norm continuous. The paper generalizes the result of \textit{R.-W. Yang} [Integral Equations Oper. Theory 28, No. 2, 238--244 (1997; Zbl 0903.47002)] which was obtained for the unilateral shift of multiplicity one.