Symmetries for J-projection

Abstract

Let $$\mathcal{B}(\mathcal{H})$$ be the algebra of all bounded linear operators on a separable complex Hilbert space $$\mathcal{H}$$. We introduce the J-decomposition property for projections in $$\mathcal{B}(\mathcal{H})$$, and prove that the projection E in $$\mathcal{B}(\mathcal{H})$$ has J-decomposition property with respect to a particular space decomposition, which is related to Hal-mos’ two projections theory. Using this, we characterize symmetries J such that the projection E is a J-projection (or J-positive projection, or J-negative projection). Also, we give the explicit representations of the maximum and the minimum of symmetries J such that the projection E is J-positive (or J-negative).