Some results on frames by pre-frame operators in Q-Hilbert spaces

Quaternionic Hilbert (Q-Hilbert) spaces are frequently used in applied physical sciences and especially in quantum physics. In order to solve some problems of many nonlinear physical systems, the frame theory of Q-Hilbert spaces was studied. Frames in Q-Hilbert spaces not only retain the frame properties, but also have some advantages, such as a simple structure for approximation. In this paper, we first characterized Hilbert (orthonormal) bases, frames, dual frames and Riesz bases, and obtained the accurate expressions of all dual frames of a given frame by taking advantage of pre-frame operators. Second, we discussed the constructions of frames with the help of the pre-frame operators and gained some more general methods to construct new frames. Moreover, we obtained a necessary and sufficient condition for the finite sum of frames to be a (tight) frame, and the obtained results further enriched and improved the frame theory of the Q-Hilbert space.