A Decomposition Theorem for the Translation-Invariant Subspace of a Canonical Differential Operator

Abstract

Let H be a unitary space of even dimension 2 m, and let y be a linear operator in H for which y* = − y, y² = − I. The orthogonal projectors P_± = 1/2. (I ± i y) define two subspaces H₊ and H₋ in H. We shall assume that H₊. = dim H₋ = m. We denote by H₀ a certain subspace in H of dimension m, whose elements satisfy the relation (y f, g) = 0, f, g ∈ H₀.