Real-linear operators on quaternionic Hilbert space

Abstract

The main result is that any continuous real-linear operator A 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 A = {A_0} + {i_1}{A_1} + {i_2}{A_2} + {i_3}{A_3} , where the A ν {A_\nu } are continuous linear operators and ( i 1 , , i 2 , i 3 ) ({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 ν {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.