Accession Number:

AD0259085

Title:

COMMUTATORS, PERTURBATIONS, AND UNITARY SPECTRA

Descriptive Note:

Corporate Author:

PURDUE RESEARCH FOUNDATION LAFAYETTE IND

Personal Author(s):

Report Date:

1961-06-01

Pagination or Media Count:

1.0

Abstract:

Let A and B denote linear operators, bounded or unbounded, on a Hilbert space H of elements x. Let x x,x12 and put A sup Ax where x 1. If A and B are bounded and if C denotes the commutator of A and B, 1.1 C AB - BA, then it is well known that 1.2 C 2 A B , and that the inequality cannot be improved by replacing the 2 by 2 - with 0. Simple examples with finite matrices A 0, B 0 and A, iB hence also C even self-adjoint show that the equality of 1.2 may hold. Part I concerns an improvement of 1.2 when B is bounded but otherwise arbitrary, A and C are bounded and selfadjoint, and C is non-negative. In Part II a related problem is considered concerning perturbations of a self-adjoint operator A. In Part III applications are given of the results of Part II to semi-normal operators, Laurent matrices, measure preserving transformations, and to what correspond to certain operators occurring in scattering theory in quantum mechanics. Author

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Distribution Statement:

APPROVED FOR PUBLIC RELEASE