<p>This work presents a rigorous characterization of continuous inner products on the Hilbert space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> of Hilbert–Schmidt operators. We begin by addressing the general framework of continuous sesquilinear forms on a Hilbert space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> and provide a characterization of all continuous inner products, by means of positive operators in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {B(H)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Next, we establish necessary and sufficient conditions for an operator in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {B}(S_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to be positive. Identifying a continuous inner product with a positive operator enables us to rigorously describe inner products on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$S_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>.</p>

Inner Products on the Hilbert Space $S_2$ of Hilbert–Schmidt Operators

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2026年3月13日

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Josué I. Rios-Cangas

摘要

This work presents a rigorous characterization of continuous inner products on the Hilbert space $S_2$ $S_{2}$ of Hilbert–Schmidt operators. We begin by addressing the general framework of continuous sesquilinear forms on a Hilbert space $\mathcal {H}$ $H$ and provide a characterization of all continuous inner products, by means of positive operators in $\mathcal {B(H)}$ $B (H)$ . Next, we establish necessary and sufficient conditions for an operator in $\mathcal {B}(S_2)$ $B (S_{2})$ to be positive. Identifying a continuous inner product with a positive operator enables us to rigorously describe inner products on $S_2$ $S_{2}$ .

Inner Products on the Hilbert Space \(S_2\) of Hilbert–Schmidt Operators

摘要