Real-linear operators on quaternionic Hilbert space

Abstract:

The main result is that any continuous real-linear operator $A$ on a quaternionic Hilbert space has a unique decomposition $A = {A_0} + {i_1}{A_1} + {i_2}{A_2} + {i_3}{A_3}$, where the ${A_\nu }$ are continuous linear operators and $({i_{1,}},{i_2},{i_3})$ is any right-handed orthonormal triad of vector quaternions. Other results concern the place of the colinear and complex-linear operators in this characterisation and the effect on the ${A_\nu }$ of a rotation of the triad of vector quaternions. A new result concerning symplectic images of a quaternionic Hilbert space is also presented.