Products of Projections, Polar Decompositions and Norms of Differences of Two Projections

Some characterizations of products of two projections on a Hilbert space are generalized to the case of products of a finite number of projections on a Hilbert C^∗-module. An example is constructed to show that in the Hilbert C^∗-module case, X and its subset X⊥ can be different, where X denotes the set of all products of two projections on a Hilbert C^∗-module, and X⊥ consists of those elements in X that have the polar decomposition. Some new phenomena are revealed when an operator is taken in X instead of X⊥. Some refinements are made on norms of differences of two projections.