Chemistry - Hilbert-Schmidt operator

In mathematics, a Hilbert-Schmidt operator is a bounded operator A on a Hilbert space H₁->H₂ such that there exists an orthonormal basis $\{e_i : i \in I\}$ of H₁ such that

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

is finite.

Let A and B are two Hilbert-Schmidt operators, the Hilbert-Schmidt inner product can be defined as

$\langle A,B \rangle_{HS} = \sum_{i \in I} \langle Ae_i, Be_i \rangle.$

This definition is independent of the choice of orthonormal basis

The Hilbert-Schmidt operators form an ideal in the algebra of bounded operators on H, which is usually not closed in the norm topology. They also form a Hilbert space, and can be shown to be isometrically isomorphic to the tensor product of Hilbert spaces $\overline{H} \otimes H$ .

Categories: Operator theory