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

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

where $\|\ \|$ is the norm of H, $\{e_{i}:i\in I\}$ an orthonormal basis of H, and Tr is the trace of a nonnegative self-adjoint operator. Note that the index set need not be countable. This definition is independent of the choice of the basis, and therefore

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

for $A_{i,j}=\langle e_{i},Ae_{j}\rangle$ and $\|A\|_{2}$ the Schatten norm of $A$ for p = 2. In Euclidean space $\|\ \|_{HS}$ is also called Frobenius norm, named for Ferdinand Georg Frobenius.

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

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

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

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

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.

See also

• Frobenius inner product