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.
A mapping is a Hilbert-Schmidt functional if it is a bounded bilinear functional.
A bounded linear mapping is weakly Hilbert-Schmidt if for all the mapping
is a Hilbert-Schmidt functional and for some real number .
- 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
- Kadison, Richard V.; Ringrose, John R. (1997), Fundamentals of the theory of operator algebras. Vol. I, Graduate Studies in Mathematics 15, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0819-1, MR 1468229 (see p. 127)
- Kadison and Ringrose, (see p. 131)