Strictly cyclic weighted shifts
Abstract: E. Nordgren recently showed that certain weighted shift operators on Hilbert space have the property that they commute with no unbounded closed densely defined linear transformations. In the present paper, it is shown that this property for a weighted shift operator is equivalent to the existence of a strictly cyclic vector for the weakly closed algebra generated by the weighted shift. After establishing some tests for the existence of such a strictly cyclic vector, several further examples are given of weighted shift operators satisfying Nordgren's commutativity property.
-  R. L. Kelley, Weighted shifts on Hilbert space, Thesis, University of Michigan, Ann Arbor, Mich., 1966.
-  Eric A. Nordgren, Closed operators commuting with a weighted shift, Proc. Amer. Math. Soc. 24 (1970), 424–428. MR 0257786, https://doi.org/10.1090/S0002-9939-1970-0257786-X
-  A. L. Shields and L. J. Wallen, The commutants of certain Hilbert space operators, Indiana Univ. Math. J. 20 (1970/1971), 777–788. MR 0287352, https://doi.org/10.1512/iumj.1971.20.20062
- R. L. Kelley, Weighted shifts on Hilbert space, Thesis, University of Michigan, Ann Arbor, Mich., 1966.
- E. Nordgren, Closed operators commuting with a weighted shift, Proc. Amer. Math. Soc. 24 (1970), 424-428. MR 0257786 (41:2435)
- A. L. Shields and L. J. Wallen, The commutant of certain Hilbert space operators (to appear). MR 0287352 (44:4558)
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47.40
Retrieve articles in all journals with MSC: 47.40
Keywords: Weighted shift operators, closed linear transformations, strictly cyclic vectors, commutant of a weighted shift
Article copyright: © Copyright 1971 American Mathematical Society