Analytic m-isometries without the wandering subspace property.

The wandering subspace problem for an analytic norm-increasing m-isometry T on a Hilbert space H asks whether every T-invariant subspace of H can be generated by a wandering subspace. An affirmative solution to this problem for m = 1 is ascribed to Beurling-Lax-Halmos, while that for 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-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. We also show that if the wandering subspace property fails for an analytic norm-increasing m-isometry, then it fails miserably in the sense that the smallest T-invariant subspace generated by the wandering subspace is of infinite codimension. [ABSTRACT FROM AUTHOR]