Characterizations and Representations of H-S-Frames in Hilbert Spaces

H-S-frame is in essence a more general operator-valued frame than generalized frames. In this paper, we aim at studying the characterizations and representations of H-S-frames in H (Hilbert space). We first introduce the notion of H-S-preframe operator, and characterize the H-S-frames, Parseval H-S-frames, H-S-Riesz bases, H-S-orthonormal bases and dual H-S-frames with the help of H-S-preframe operators, and obtain the accurate expressions of all dual H-S-frames of a given H-S-frame by drawing support from H-S-preframe operators. Then we discuss the sum of H-S-frames through the properties of H-S-preframe operators. Finally, with the help of the approaches and skills of frame theory, we present the representations of H-S-frames and H-S-Bessel sequences. Specifically, the necessary and sufficient condition for the H-S-frame to be represented as a combination of two H-S-orthonormal bases is that the H-S-frame is an H-S-Riesz basis.