DIFFERENCE AND SIMILARITY MODELS OF 2 IDEMPOTENT OPERATORS

Abstract

In a recent work, Hartwig and Putcha obtained a complete characterization of those finite matrices which can be expressed as the difference of two idempotents. Extending this result to operators on a possibly infinite-dimensional Hilbert space seems more difficult. In this paper, we initiate its study and obtain, among other things, (1) that not every nilpotent operator is the difference of two idempotents, (2) that if T is the difference of two idempotents, then the spectra of T and -T differ at most by the two points +/-1, and (3) a characterization of differences of two idempotents among normal operators. In the second part of the paper, we develop some similarity-invariant models of two idempotents. These are analogous to the known unitary-equivalence-invariant models for two orthogonal projections.