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<p>If <i>K</i> is a bounded operator on a separable Hilbert space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \mathcal {H} \)</EquationSource> </InlineEquation>, we will characterize scalable <i>K</i>-frames with respect to positive diagonal operators on the sequence space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( \ell ^2 \)</EquationSource> </InlineEquation>. This is the simplest method to generate a Parseval <i>K</i>-frame without requiring the invertibility of frame operator. One first goal is to modify the content of Theorem 3.8 of Ramesan (Mat Vesn 75:225–234, 2023) (and Theorem 4.1 of Ramesan (Palest J Math 12:493–500, 2023)). First, we provide a counterexample to its invalidity and then we prove it with an additional assumption. We also investigate the stability of scalable <i>K</i>-frames under suitable operators.</p>

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Scalable K-frames and Their Stability Under Some Operators

  • 期刊论文

  • 发表日期：

    2026年1月3日

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  • Afsaneh Mahamed,
  • S. Mohammad Moshtaghioun,
  • Ahmad Ahmadi

摘要

If K is a bounded operator on a separable Hilbert space \( \mathcal {H} \) , we will characterize scalable K-frames with respect to positive diagonal operators on the sequence space \( \ell ^2 \) . This is the simplest method to generate a Parseval K-frame without requiring the invertibility of frame operator. One first goal is to modify the content of Theorem 3.8 of Ramesan (Mat Vesn 75:225–234, 2023) (and Theorem 4.1 of Ramesan (Palest J Math 12:493–500, 2023)). First, we provide a counterexample to its invalidity and then we prove it with an additional assumption. We also investigate the stability of scalable K-frames under suitable operators.