Hilbert-Schmidt operator
An operator acting on a Hilbert space
such that for any orthonormal basis
in
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 for which the condition
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applies to its -numbers
and its eigen values
; here
is a trace-class operator (
is the adjoint of
and
is the trace of an operator
). The set of all Hilbert–Schmidt operators on a fixed space
forms a Hilbert space with scalar product
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If is the resolvent of
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 (which explains the origin of the name).
Comments
The -numbers or singular values of
are the (positive) eigen values of the self-adjoint operator
. Instead of Hilbert–Schmidt operator one also says "of Hilbert–Schmidt class operatorHilbert–Schmidt class" . A bounded operator
on a Hilbert space is said to be of trace class if
for arbitrary complete orthonormal systems
,
. Equivalently,
is of trace class if
. The trace of such an operator is defined as
, where
is any orthonormal basis. The product of two Hilbert–Schmidt operators is of trace class and the converse is also true.
The norm in the above article is not the usual operator norm of
but its Hilbert–Schmidt norm.
References
[a1] | M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972) |
[a2] | I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1977) |
[a3] | N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian) |
Hilbert-Schmidt operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert-Schmidt_operator&oldid=15046