View source for Hilbert-Schmidt operator

An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473501.png" /> acting on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473502.png" /> such that for any orthonormal basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473503.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473504.png" /> the following condition is met:

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(however, this need be true for some basis only). A Hilbert–Schmidt operator is a [[Compact operator|compact operator]] for which the condition

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applies to its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473507.png" />-numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473508.png" /> and its eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473509.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735010.png" /> is a trace-class operator (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735011.png" /> is the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735013.png" /> is the trace of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735014.png" />). The set of all Hilbert–Schmidt operators on a fixed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735015.png" /> forms a Hilbert space with scalar product

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735017.png" /> is the resolvent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735018.png" /> and

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is its regularized characteristic determinant, then the Carleman inequality

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holds.

A typical representative of a Hilbert–Schmidt operator is a [[Hilbert–Schmidt integral operator|Hilbert–Schmidt integral operator]] (which explains the origin of the name).



====Comments====
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735022.png" />-numbers or [[singular value]]s of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735023.png" /> are the (positive) eigen values of the self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735024.png" />. Instead of Hilbert–Schmidt operator one also says  "operator of Hilbert–Schmidt class" . A bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735025.png" /> on a Hilbert space is said to be of trace class if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735026.png" /> for arbitrary complete orthonormal systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735028.png" />. Equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735029.png" /> is of trace class if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735030.png" />. The trace of such an operator is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735032.png" /> is any orthonormal basis. The product of two Hilbert–Schmidt operators is of trace class and the converse is also true.

The norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735033.png" /> in the above article is not the usual operator norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735034.png" /> but its [[Hilbert–Schmidt norm|Hilbert–Schmidt norm]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Reed,   B. Simon,   "Methods of modern mathematical physics" , '''1. Functional analysis''' , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.C. Gohberg,   S. Goldberg,   "Basic operator theory" , Birkhäuser  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.I. Akhiezer,   I.M. Glazman,   "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR></table>