Some Results on Oblique Dual and Approximate Oblique Dual Hilbert-Schmidt Frames

The concept of Hilbert-Schmidt frames (HS-frames) is more general than that of g-frames. In this paper, we investigate the oblique dual (OD) and epsilon-approximate oblique dual (epsilon-AOD) HS-frames on closed subspaces W and V of a separable Hilbert space H. We first introduce the concepts of OD and epsilon-AOD HS-frames. Then we present some conditions for a pair of OD and epsilon-AOD HS-frames to be symmetric. We also obtain the algebraic formula for all OD and epsilon-AOD HS-frames on V for a given HS-frame in W. As application, we get the canonical OD HS-frame has a minimum norm among all the representations of elements in W. We also discuss the perturbations of epsilon-AOD HS-frames. Finally, applying our results, we not only recover some known results but also derive a new result in the classical Hilbert space frame setting.