<p>We give a characterization of Hilbert-Schmidt frames for the Hilbert-Schmidt class <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_948_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {C}_2(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">C</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of surjectivity of a bounded operator on <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_948_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {C}_2(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">C</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. A difference between standard frames in separable Hilbert spaces and Hilbert-Schmidt frames for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_948_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {C}_2(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">C</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is discussed. Hilbert-Schmidt frames for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_948_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {C}_2(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">C</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are constructed using bounded operators on the Hilbert spaces <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_948_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {C}_2(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">C</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_948_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> that satisfy certain properties.</p>

A note on Hilbert-Schmidt frames for $\mathcal {C}_2(\mathcal {H})$

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发表日期：

2025年7月28日

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Jyoti

摘要

We give a characterization of Hilbert-Schmidt frames for the Hilbert-Schmidt class $\mathcal {C}_2(\mathcal {H})$ $C_{2} (H)$ in terms of surjectivity of a bounded operator on $\mathcal {C}_2(\mathcal {H})$ $C_{2} (H)$ . A difference between standard frames in separable Hilbert spaces and Hilbert-Schmidt frames for $\mathcal {C}_2(\mathcal {H})$ $C_{2} (H)$ is discussed. Hilbert-Schmidt frames for $\mathcal {C}_2(\mathcal {H})$ $C_{2} (H)$ are constructed using bounded operators on the Hilbert spaces $\mathcal {C}_2(\mathcal {H})$ $C_{2} (H)$ and $\mathcal {H}$ $H$ that satisfy certain properties.

A note on Hilbert-Schmidt frames for \(\mathcal {C}_2(\mathcal {H})\)

摘要