Complex structure on a real Hilbert space and symplectic structure on a complex Hilbert space

Abstract

Alternative ways of complexifying a real Hilbert space and quaternionizing a complex Hilbert space are described. The work gives some insight into why even though in the finite-dimensional case a complex Hilbert space when viewed as a real Hilbert space and a quaternionic Hilbert space when viewed as a complex Hilbert space have twice their original dimensions, the degrees of freedom of the linear operators remain unchanged. Many ramifications are discussed, among them the reconciliation of the linearity of the adjoint of a semilinear (antilinear) map from one complex Hilbert space to another with the semilinearity (antilinearity) of the adjoint of a semilinear (antilinear) map from one complex Hilbert space to itself. Groundwork is prepared for the study of the noncommutative algebra of additive operators on a quaternionic Hilbert space.