Hilbert–Schmidt operator

In mathematics, a Hilbert–Schmidt operator is a bounded operator "A" on a Hilbert space "H" with finite Hilbert–Schmidt norm, meaning that there exists an orthonormal basis $\{e\_i\; :\; i\; in\; I\}$ of "H" with the property

:

If this is true for one orthonormal basis, it is true for any other orthonormal basis.

Let "A" and "B" be two Hilbert–Schmidt operators. The Hilbert–Schmidt inner product can be defined as

:$langle\; A,B\; angle\_mathrm\{HS\}\; =\; operatorname\{trace\}\; A^tB=\; sum\_\{i\; in\; I\}\; langle\; Ae\_i,\; Be\_i\; angle.$The induced norm is called the Hilbert–Schmidt norm::$lVert\; A\; Vert\_mathrm\{HS\}^2\; =\; sum\_\{i\; in\; I\}\; lVert\; Ae\_i\; Vert^2.$

This definition is independent of the choice of orthonormal basis, and is analogous to the Frobenius norm for operators on a finite-dimensional vector space.

The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on "H". The Hilbert–Schmidt operators are closed in the norm topology if, and only if, "H" is finite dimensional. They also form a Hilbert space, and 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".

ee also

* Nuclear operator
* Trace class
* Mertens theorem
* Hilbert-Schmidt integral operator