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 and $\{e_i : i\in I\}$ an orthonormal basis of H for an index set $I$.[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 $A$. 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.

Functionals

A mapping $\phi:H_1\times H_2\to C$ is a Hilbert-Schmidt functional if it is a bounded bilinear functional.[3]

A bounded linear mapping $L:H_1\times H_2\to K$ is weakly Hilbert-Schmidt if for all $v\in K$ the mapping

$\phi_v = (u_1,u_2)\mapsto\langle L(u_1,u_2), v\rangle$

is a Hilbert-Schmidt functional and $\|\phi_v\|\leq M\|v\|$ for some real number $M\geq 0$.[4]

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
3. ^ 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)
4. ^ Kadison and Ringrose, (see p. 131)