Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-17T17:15:09.801Z Has data issue: false hasContentIssue false

Unitary equivalence and reductibility or invertibly weighted shifts

Published online by Cambridge University Press:  17 April 2009

Alan Lambert
Affiliation:
University of Kentucky, Lexington, Kentucky, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let H be a complex Hilbert space and let {A1, A2, …} be a uniformly bounded sequence of invertible operators on H. The operator S on l2(H) = HH ⊕ … given by Sx0, x1, …〉 = 〈0, A1x0, A2x1, …〉 is called the invertibly veighted shift on l2(H) with weight sequence {An }. A matricial description of the commutant of S is established and it is shown that S is unitarily equivalent to an invertibly weighted shift with positive weights. After establishing criteria for the reducibility of S the following result is proved: Let {B1, B2, …} be any sequence of operators on an infinite dimensional Hilbert space K. Then there is an operator T on K such that the lattice of reducing subspaces of T is isomorphic to the corresponding lattice of the W* algebra generated by {B1, B2, …}. Necessary and sufficient conditions are given for S to be completely reducible to scalar weighted shifts.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Fillmore, P.A., Notes on operator theory (Van Nostrand Reinhold Math. Studies, No. 30. Van Nostrand Reinhold, New York, 1970).Google Scholar
[2]Halmos, Paul R., “Shifts on Hilbert spaces”, J. reine angew. Math. 208 (1961), 102112.CrossRefGoogle Scholar
[3]Halmos, Paul R., A Hilbert space problem book (Van Nostrand, Princeton, New Jersey; Toronto, Ontario; London; 1967).Google Scholar
[4]Kelley, Robert Lee, “Weighted shifts on Hilbert space”, (doctoral thesis, University of Michigan, Ann Arbor, 1966).Google Scholar
[5]Shields, A.L. and Wallen, L.J., “The commutant of certain Hilbert space operators”, (to appear).Google Scholar