Compact operators as products of projections

Abstract

Products of projections appear in the analysis of many algorithms. Indeed, many algorithmsare merely products of projections iterated. For example, the best approximation of a point from the intersection of a set of subspaces can be determined from the following parallel algorithm. Define x n + 1 = 1 k ∑ i = 0 k P i where P i is the orthogonal projection onto a subspace C i . Then lim n→∞ ( x n ) is the best approximation to x 0 in the intersection of the C i . This algorithm can be represented as a product of projections in a higher dimensional space. When forming hypotheses about a product of projections, the natural question that arises is “Which linear operators can be factored into a product of projections?” This note demonstrates that many bounded linear operators can be represented as scalar mutliples of a product of projections. In the case of compact operators with norm strictly less than one, a constructive method for factoring the operator can often be obtained. Furthermore, all compact operators with norm strictly less than one have a simple extension in a higher dimensional Hilbert space and can be represented as a product of projections. Of course, this implies that EVERY compact operator has an extension that is a scalar times a product of projections.