2018/11/29 by Akash Anand, Sameer Chavan, Anand, Akash +3 · 1 citation
Mathematics · #05C20 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Primary 47B37 #Secondary 47A15 #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1811.12080
openalex publication_date 2018/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.