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Canonical Dual \(\varvec{K}\)-g-Bessel Sequences and \(\varvec{K}\)-g-Frame Sequences

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Abstract

The classical canonical dual for a K-g-frame is absent since the frame operator may not be invertible. We introduce the concept of canonical dual K-g-Bessel sequences, which have some properties similar to classical canonical dual, to investigate the duality of K-g-frames. We obtain the exact form of the canonical dual K-g-Bessel sequences for Parseval K-g-frames and some other related results. We also introduce the concept of K-g-frame sequences to explore the properties of local K-g-frames. We present a necessary and sufficient condition under which a K-g-frame sequence is a K-g-frame, and necessary and sufficient conditions for a sequence of bounded operators to be a K-g-frame sequence. We end the paper by examining the relationship between K-g-frame sequences and g-frames for closed subspaces, and the relationship between two K-g-frame sequences and the ranges of the involved operators.

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Acknowledgements

The author thanks the anonymous referees for valuable suggestions and comments which have led to a significant improvement of this article.

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Correspondence to Zhong-Qi Xiang.

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The research is supported by the National Natural Science Foundation of China (Nos. 11761057 and 11561057), the Natural Science Foundation of Jiangxi Province, China (No. 20151BAB201007), and the Science Foundation of Jiangxi Education Department (No. GJJ151061).

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Xiang, ZQ. Canonical Dual \(\varvec{K}\)-g-Bessel Sequences and \(\varvec{K}\)-g-Frame Sequences. Results Math 73, 17 (2018). https://doi.org/10.1007/s00025-018-0776-y

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