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Hilbert–Schmidt operators : ウィキペディア英語版
Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named for David Hilbert and Erhard Schmidt, is a bounded operator ''A'' on a Hilbert space ''H'' with finite Hilbert–Schmidt norm
:\|A\|^2_= (A^=\sum_ |A_|^2 = \|A\|^2_2
for A_=\langle e_i, Ae_j \rangle and \|A\|_2 the Schatten norm of A for ''p=2''. In Euclidean space \|\ \|_ is also called Frobenius norm, named for Ferdinand Georg Frobenius.
The product of two Hilbert–Schmidt operators has finite trace class norm; therefore, if ''A'' and ''B'' are two Hilbert–Schmidt operators, the Hilbert–Schmidt inner product can be defined as
:\langle A,B \rangle_\mathrm = \operatorname (A^
*B)
= \sum_ \langle Ae_i, Be_i \rangle.
The Hilbert–Schmidt operators form a two-sided
*-ideal
in the Banach algebra of bounded operators on ''H''. They also form a Hilbert space, which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces
:H^
* \otimes H, \,
where ''H
*'' is the dual space of ''H''.
The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, ''H'' is finite-dimensional.
An important class of examples is provided by Hilbert–Schmidt integral operators.
Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact.
==References==


抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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