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Analytic m-isometries without the wandering subspace property.
- Source :
-
Proceedings of the American Mathematical Society . May2020, Vol. 148 Issue 5, p2129-2142. 14p. - Publication Year :
- 2020
-
Abstract
- 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]
- Subjects :
- *DIRECTED graphs
*HILBERT space
*SUBSPACES (Mathematics)
Subjects
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 148
- Issue :
- 5
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 142355085
- Full Text :
- https://doi.org/10.1090/proc/14894