# 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