Nonsymmetric projections in Hilbert space

Abstract

O Introduction. An initial investigation into the kind of operators which can be obtained as the difference of two projections led to the study presented below. In this paper a characterization is given for the general (not necessarily symmetric) bounded linear idempotent operator, or projection, on Hubert space. These results are applied to the investigation of a projection problem and to a " weak" ordering of such operators. The paper falls naturally into two parts. In the first we give two theorems and several more or less direct consequences which together provide the characterization. In the second part we apply these results to the investigation and solution of a problem which is of importance in probability and statistics. A sketch of the role of this problem in statistical theory and an examination of how our results fit in with previous conclusions complete the present study.