Hilbert–Schmidt operator

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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

$\sum_{i\in I} \|Ae_i\|^2 < \infty.$

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

$\langle A,B \rangle_\mathrm{HS} = \operatorname{trace} A^tB = \sum_{i \in I} \langle Ae_i, Be_i \rangle.$

The induced norm is called the Hilbert–Schmidt norm:

$\lVert A \rVert_\mathrm{HS}^2 = \sum_{i \in I} \lVert Ae_i \rVert^2.$

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

$H^* \otimes H,$