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Generalized Dual Hilbert–Schmidt Frames and Their Topological Properties

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Abstract

This paper addresses the Hilbert–Schmidt frame (HS-frame) theory. We introduce the concept of generalized dual HS-frame (g-dual HS-frame) which generalizes that of g-dual frame. We prove that two equivalent HS-frames form a g-dual HS-frame pair, characterize operators on \(\ell ^2\) that transform a pair of HS-Riesz bases into a g-dual HS-frame pair, and present a parametric expression of all g-dual HS-frames of an arbitrarily given HS-frame. Also the perturbation-stability and topological properties of g-dual HS-frames are investigated. Finally, applying our results, we not only recover some known results but also derive some new results in the classical Hilbert space frame setting.

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Funding

This work was supported by National Natural Science Foundation of China (Grant No. 11971043).

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  1. School of Mathematics,, Statistics and Mechanics, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Rui-Qi Dong & Yun-Zhang Li

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  1. Rui-Qi Dong
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  2. Yun-Zhang Li
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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Correspondence to Yun-Zhang Li.

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Supported by National Natural Science Foundation of China (Grant Nos. 12371091, 11971043).

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Dong, RQ., Li, YZ. Generalized Dual Hilbert–Schmidt Frames and Their Topological Properties. Results Math 79, 80 (2024). https://doi.org/10.1007/s00025-023-02110-2

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