For a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-finite and countably generated measure space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(\Omega , \mathcal {A}, \mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="script">A</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we study the matrix-valued frame of the matrix-valued function space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^2(\Omega , \mathbb {C}^{m\times n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where the lower frame condition depends on a bounded linear operator <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Θ</mi> </math></EquationSource> </InlineEquation> acting on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^2(\Omega , \mathbb {C}^{m\times n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and call it a <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Θ</mi> </math></EquationSource> </InlineEquation>-M-frame. This is inspired by the work of Gǎvruta for discrete frames in separable Hilbert spaces. Firstly, we give a characterization of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Θ</mi> </math></EquationSource> </InlineEquation>-M-frames. Necessary and sufficient conditions for the existence of a <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Θ</mi> </math></EquationSource> </InlineEquation>-M-frame in terms of Gǎvruta-type atomic systems are given. Linear preservers for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Θ</mi> </math></EquationSource> </InlineEquation>-M-frames, that is, bounded linear maps that preserve both the frame conditions, are given. Finally, we give a sufficient condition for the Paley-Wiener type perturbation for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Θ</mi> </math></EquationSource> </InlineEquation>-M-frames.

Gǎvruta-type matrix-valued frames over measure spaces

期刊论文
发表日期：

2026年7月30日

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Khalil Ahmad,
Shyam Lal,
Mohammed Jamali,
Pankaj Sharma

摘要

For a $\sigma $ $σ$ -finite and countably generated measure space $(\Omega , \mathcal {A}, \mu )$ $(Ω, A, μ)$ , we study the matrix-valued frame of the matrix-valued function space $L^2(\Omega , \mathbb {C}^{m\times n})$ $L^{2} (Ω, C^{m \times n})$ , where the lower frame condition depends on a bounded linear operator $\Theta $ $Θ$ acting on $L^2(\Omega , \mathbb {C}^{m\times n})$ $L^{2} (Ω, C^{m \times n})$ and call it a $\Theta $ $Θ$ -M-frame. This is inspired by the work of Gǎvruta for discrete frames in separable Hilbert spaces. Firstly, we give a characterization of $\Theta $ $Θ$ -M-frames. Necessary and sufficient conditions for the existence of a $\Theta $ $Θ$ -M-frame in terms of Gǎvruta-type atomic systems are given. Linear preservers for $\Theta $ $Θ$ -M-frames, that is, bounded linear maps that preserve both the frame conditions, are given. Finally, we give a sufficient condition for the Paley-Wiener type perturbation for $\Theta $ $Θ$ -M-frames.