Analytic 𝑚-isometries without the wandering subspace property

Abstract

The wandering subspace problem for an analytic norm-increasing m m -isometry T T on a Hilbert space H \mathcal {H} asks whether every T T -invariant subspace of H \mathcal {H} can be generated by a wandering subspace. An affirmative solution to this problem for m = 1 m=1 is ascribed to Beurling-Lax-Halmos, while that for m = 2 m=2 is due to Richter. In this paper, we capitalize on the idea of weighted shift on a one-circuit directed graph to construct a family of analytic cyclic 3 3 -isometries which do not admit the wandering subspace property and which are norm-increasing on the orthogonal complement of a one-dimensional space. Further, on this one-dimensional space, their norms can be made arbitrarily close to 1 1 . We also show that if the wandering subspace property fails for an analytic norm-increasing m m -isometry, then it fails miserably in the sense that the smallest T T -invariant subspace generated by the wandering subspace is of infinite codimension.