Accession Number : AD0259085

Title :   COMMUTATORS, PERTURBATIONS, AND UNITARY SPECTRA

Corporate Author : PURDUE RESEARCH FOUNDATION LAFAYETTE IND

Personal Author(s) : PUTNAM,C.R.

Report Date : JUN 1961

Pagination or Media Count : 1

Abstract : Let A and B denote linear operators, bounded or unbounded, on a Hilbert space H of elements x. Let x = (x,x)1/2 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)

Descriptors :   *OPERATORS (MATHEMATICS), *PERTURBATION THEORY, ALGEBRAS, FUNCTIONAL ANALYSIS, MATRICES(MATHEMATICS), MEASURE THEORY, QUANTUM THEORY, TOPOLOGY

Distribution Statement : APPROVED FOR PUBLIC RELEASE