Scalar Identities for Pairs of Orthogonal Projections in Hilbert Spaces

Abstract

Let M and N be closed subspaces of a Hilbert space <inline-formula> <tex-math notation="LaTeX">$\mathbb {H}$ </tex-math></inline-formula>, and let <inline-formula> <tex-math notation="LaTeX">$\Pi _{M}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\Pi _{N}$ </tex-math></inline-formula> denote the corresponding orthogonal projections. In this paper, we study pairs of projections that satisfy the scalar identity <inline-formula> <tex-math notation="LaTeX">$\Pi _{M} \Pi _{N} \Pi _{M} = \gamma \Pi _{M}$ </tex-math></inline-formula> for some <inline-formula> <tex-math notation="LaTeX">$\gamma \in (0,1$ </tex-math></inline-formula>]. We establish several equivalent formulations of this property using block operator matrices, angles between subspaces, and extremal norm equalities. In addition, we compute explicit formulas for the norms of the sum and the anticommutator of such projections. We also show that this scalar condition implies the pair of projections is acute, and we prove that the property is preserved under subprojections.