Some Properties of Canonical Dual K-Bessel Sequences for Parseval K-Frames

• Main Author: Zhong-Qi Xiang
• Format: Article
• Language: English
• Published: Hindawi Limited 2019-01-01
• Series: Journal of Function Spaces
• Online Access: http://dx.doi.org/10.1155/2019/9862369

The concept of canonical dual K-Bessel sequences was recently introduced, a deep study of which is helpful in further developing and enriching the duality theory of K-frames. In this paper we pay attention to investigating the structure of the canonical dual K-Bessel sequence of a Parseval K-frame and some derived properties. We present the exact form of the canonical dual K-Bessel sequence of a Parseval K-frame, and a necessary and sufficient condition for a dual K-Bessel sequence of a given Parseval K-frame to be the canonical dual K-Bessel sequence is investigated. We also give a necessary and sufficient condition for a Parseval K-frame to have a unique dual K-Bessel sequence and equivalently characterize the condition under which the canonical dual K-Bessel sequence of a Parseval K-frame admits a unique dual K⁎-Bessel sequence. Finally, we obtain a minimal norm property on expansion coefficients of elements in the range of K resulting from the canonical dual K-Bessel sequence of a Parseval K-frame.