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Hilbert–Schmidt operator
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{{Short description|Topic in mathematics}} In [[mathematics]], a '''Hilbert–Schmidt operator''', named after [[David Hilbert]] and [[Erhard Schmidt]], is a [[bounded operator]] <math> A \colon H \to H </math> that acts on a [[Hilbert space]] <math> H </math> and has finite '''Hilbert–Schmidt norm''' <math display="block">\|A\|^2_{\operatorname{HS}} \ \stackrel{\text{def}}{=}\ \sum_{i \in I} \|Ae_i\|^2_H,</math> where <math>\{e_i: i \in I\}</math> is an [[orthonormal basis]].<ref name="MathWorld">{{cite web |last=Moslehian |first=M. S. |title=Hilbert–Schmidt Operator (From MathWorld) |url=http://mathworld.wolfram.com/Hilbert-SchmidtOperator.html}}</ref><ref name="EOM">{{eom |first=M. I. |last=Voitsekhovskii |title=Hilbert-Schmidt operator}}</ref> The index set <math> I </math> need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning.{{sfn | Schaefer | 1999 | p=177}} This definition is independent of the choice of the orthonormal basis. In finite-dimensional [[Euclidean space]], the Hilbert–Schmidt norm <math>\|\cdot\|_\text{HS}</math> is identical to the [[matrix norm#Frobenius norm|Frobenius norm]].
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