# 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**

where is the norm of *H*, 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

for and the Schatten norm of for *p* = 2. In Euclidean space 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

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

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

## References

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