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.
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