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

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

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.^[1][2] Note that the index set need not be countable. This definition is independent of the choice of the basis, and therefore

$\|A\|^2_{HS}=\sum_{i,j} |A_{i,j}|^2 = \|A\|^2_2$

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.

References

1. Moslehian, M.S. "Hilbert–Schmidt Operator (From MathWorld)".
2. Voitsekhovskii, M.I. (2001), "H/h047350", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4