Strong Projections in Hilbert Space and Quantum Logic

Abstract

In the paper we study linear operators on complex Hilbert spaces which are strong real-orthogonal projections. It is a generalization of such standard (complex) orthogonal projections for which only the real part of scalar product vanishes. We compare order properties of the orthogonal and of the strong real-orthogonal projections. We prove that the set of all strong real-orthogonal the projections in the complex space is the quantum pseudo logic. We also prove an analogy of Gleason’s theorem.