
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 selfadjoint 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 nonnegative. In Part II a related problem is considered concerning perturbations of a selfadjoint operator A. In Part III applications are given of the results of Part II to seminormal 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