Analytic $m$-isometries without the wandering subspace property

The wandering subspace problem for an analytic norm-increasing $m$-isometry $T$ on a Hilbert space $\mathcal H$ asks whether every $T$-invariant subspace of $\mathcal H$ 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 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.