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Unitary equivalence and reductibility or invertibly weighted shifts

Published online by Cambridge University Press:  17 April 2009

Alan Lambert
Alan Lambert
Affiliation:
University of Kentucky, Lexington, Kentucky, USA.
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Abstract

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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) = H ⊕ H ⊕ … given by S〈x0, 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
Information
Bulletin of the Australian Mathematical Society , Volume 5 , Issue 2 , October 1971 , pp. 157 - 173
Copyright
Copyright © Australian Mathematical Society 1971

References

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