Extensions of intertwining relations (Q1122757)

Let \(H_ 1\) and \(H_ 2\) be Hilbert spaces and let \(U_ 1\) and \(U_ 2\) be bounded linear operators on \(H_ 1\) and \(H_ 2\), respectively. Suppose that \(M_ 1\) is a closed subspace of \(H_ 1\) invariant under \(U_ 1\) and that X: \(M_ 1\to H_ 2\) is a bounded linear operator intertwining \(U_ 2\) and \(U_ 1| M_ 1\), i.e., \(X(U_ 1| M_ 1)=U_ 2X\). The authors pose a problem: Find a bounded linear operator Y: \(H_ 1\to H_ 2\) which extends X and satisfies the intertwining equation \(YU_ 1=U_ 2Y\). The main result of this paper is a necessary and sufficient condition for the existence of solutions in the case where \(U_ 1\) and \(U_ 2\) are isometries. It is shown that the problem must have a solution if one of the following conditions holds: (1) \(U_ 1\) is an isometry and \(U_ 2\) is a unitary operator, (2) \(U_ 1\) and \(U_ 2\) are coisometries. As a corollary, a conjecture posed by \textit{L. B. Page} [Pac. J. Math. 36, 787-794 (1971; Zbl 0218.47006)] and concerned with the intertwining equation for a unilateral shift is answered in the affirmative.