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<p>The concept of Hilbert-Schmidt frame (HS-frame) was first proposed by Sadeghi and Arefijamaal, which is more general than that of g-frame. In this paper, we introduce the concept of Hilbert-Schmidt biframe (HS-biframe) which generalizes biframe in Hilbert spaces and Hilbert-Schmidt dual frame. We give some properties of HS-biframes, prove that an arbitrary HS-biBessel sequence can be extended to be a Parseval HS-biframe, and the HS-biframe property is possibly preserved under small perturbation, characterize bounded operators that transform a given HS-biframe to another one, and present some parametric expressions of all HS-biframes consisting of HS-Bessel sequences. Finally, using these results we not only recover some known results but also derive some new results in the classical Hilbert space biframe setting.</p>

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Properties and constructions of Hilbert-Schmidt biframes

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • Yun-Zhang Li,
  • Rui-Qi Dong

摘要

The concept of Hilbert-Schmidt frame (HS-frame) was first proposed by Sadeghi and Arefijamaal, which is more general than that of g-frame. In this paper, we introduce the concept of Hilbert-Schmidt biframe (HS-biframe) which generalizes biframe in Hilbert spaces and Hilbert-Schmidt dual frame. We give some properties of HS-biframes, prove that an arbitrary HS-biBessel sequence can be extended to be a Parseval HS-biframe, and the HS-biframe property is possibly preserved under small perturbation, characterize bounded operators that transform a given HS-biframe to another one, and present some parametric expressions of all HS-biframes consisting of HS-Bessel sequences. Finally, using these results we not only recover some known results but also derive some new results in the classical Hilbert space biframe setting.