# Hilbert–Schmidt operator

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

${\displaystyle \|A\|_{HS}^{2}={\rm {Tr}}|(A^{{}^{*}}A)|:=\sum _{i\in I}\|Ae_{i}\|^{2}}$

where ${\displaystyle \|\ \|}$ is the norm of H and ${\displaystyle \{e_{i}:i\in I\}}$ an orthonormal basis of H for an index set ${\displaystyle I}$.[1][2] Note that the index set need not be countable. This definition is independent of the choice of the basis, and therefore

${\displaystyle \|A\|_{HS}^{2}=\sum _{i,j}|A_{i,j}|^{2}=\|A\|_{2}^{2}}$

The product of two Hilbert–Schmidt operators has finite trace class norm; therefore, if A and B are two Hilbert–Schmidt operators, the Hilbert–Schmidt inner product can be defined as

${\displaystyle \langle A,B\rangle _{\mathrm {HS} }=\operatorname {tr} (A^{*}B)=\sum _{i}\langle Ae_{i},Be_{i}\rangle .}$

The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on H. They also form a Hilbert space, which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces

${\displaystyle H^{*}\otimes H,\,}$

where H* is the dual space of H.

The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, H is finite-dimensional.

An important class of examples is provided by Hilbert–Schmidt integral operators.

Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact.