Operators on hilbert space Definition and 26 Threads

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




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A

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HS


2


=

∑

i
∈
I


‖
A

e

i



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2


,


{\displaystyle \|A\|_{\operatorname {HS} }^{2}=\sum _{i\in I}\|Ae_{i}\|^{2},}
where



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⋅
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{\displaystyle \|\cdot \|}
is the norm of H,



{

e

i


:
i
∈
I
}


{\displaystyle \{e_{i}:i\in I\}}
an orthonormal basis of H. Note that the index set need not be countable; however, at most countably many terms will be non-zero. These definitions are independent of the choice of the basis.
In finite-dimensional Euclidean space, the Hilbert–Schmidt norm



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⋅

‖

HS




{\displaystyle \|\cdot \|_{\text{HS}}}
is identical to the Frobenius norm.

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