# What is Operators on hilbert space: Definition and 25 Discussions

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

HS

2

=

i

I

A

e

i

2

,

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

{\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

HS

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

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