Products of positive operators

Abstract

A new, very simple proof is given of a result of P. Y. Wu which asserts that every unitary operator on an infinite-dimensional Hilbert space is a product of positive operators. A number of mathematicians have considered the problem of writing an operator as a product of nice operators, such as positive, hermitian or normal operators. Our principal reference for this is a paper of P. Y. Wu [6], but see also [2] and [5]. This kind of question, and related questions, have also been considered in a C*-algebra context, see [3]. A core result of Wu's paper is his theorem that a unitary operator on an infinitedimensional Hilbert space is a product of (sixteen) positive operators. This is an unexpected result, given what occurs in finite dimensions. For, in the latter situation, if we make the usual identification of an operator with a finite square matrix, a theorem of C. S. Ballantine [1] asserts that a matrix is a product of positive matrices precisely when its determinant is non-negative. The aim of this paper is to present a new proof of Wu's unitary result that is simpler and more perspicuous. Our version is so simple that we do not even have to use Ballantine's theorem (Wu's proof of his unitary result does make use of this theorem). In fact, we derive a weak version of Ballantine's theorem from our methods that has the merit of also being very simple and perspicuous. 1. Lemma. Let u be a unitary element of a unital C*-algebra A. Then the matrix