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On Differences of Unitarily Equivalent Self-Adjoint Operators†

Published online by Cambridge University Press:  18 May 2009

C. R. Putnam
C. R. Putnam
Affiliation:
Purdue University, Lafayette, Indiana, U.S.A.
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1. All operators considered in this paper are bounded operators on a Hilbert space. In case A and B are self-adjoint, certain conditions on A, B and their difference

assuring the unitary equivalence of Aand B,

have recently been obtained by Rosenblum [6] and Kato [2]. The present paper will consider the problem of investigating consequences of an assumed relation of type (2) for some unitary U together with an additional hypothesis that the difference H of (1) be non-negative, so that

First, it is easy to see that if only (2) and (3) are assumed, thereby allowing H = 0, relation (2) can hold for A arbitrary with U = I (identity) and B = A. If H = 0 in (3) is not allowed, however (an impossible assumption in the finite dimensional case, incidentally, since then the trace of H is zero and hence H = 0), it will be shown, among other things, that any unitary operator U for which (2) and (3) hold must have a spectrum with a positive measure (as a consequence of (i) of Theorem 2 below). Moreover A (hence B) cannot differ from a completely continuous operator by a constant multiple of the identity (Theorem 1). In case 0 is not in the point spectrum of H, then U is even absolutely continuous (see (iv) of Theorem 2). In § 4, applications to semi-normal operators will be given.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 4 , Issue 3 , January 1960 , pp. 103 - 107
Copyright
Copyright © Glasgow Mathematical Journal Trust 1960

References

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