Some Properties of Operator Valued Frames in Quaternionic Hilbert Spaces

Quaternionic Hilbert spaces play an important role in applied physical sciences especially in quantum physics. In this paper, the operator valued frames on quaternionic Hilbert spaces are introduced and studied. In terms of a class of partial isometries in the quaternionic Hilbert spaces, a parametrization of Parseval operator valued frames is obtained. We extend to operator valued frames many of the properties of vector frames on quaternionic Hilbert spaces in the process. Moreover, we show that all the operator valued frames can be obtained from a single operator valued frame. Finally, several results for operator valued frames concerning duality, similarity of such frames on quaternionic Hilbert spaces are presented.

Automated summary

This paper introduces and studies operator valued frames on quaternionic Hilbert spaces. A parametrization of Parseval operator valued frames is obtained using a class of partial isometries in the quaternionic Hilbert spaces. Many properties of vector frames on quaternionic Hilbert spaces are extended to operator valued frames. Moreover, the paper shows that all the operator valued frames can be obtained from a single operator valued frame. Finally, several results regarding duality and similarity of operator valued frames on quaternionic Hilbert spaces are presented.