In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator
that acts on a Hilbert space
and has finite Hilbert–Schmidt norm
{\displaystyle \|A\|_{\operatorname {HS} }^{2}\ {\stackrel {\text{def}}{=}}\ \sum _{i\in I}\|Ae_{i}\|_{H}^{2},}
is an orthonormal basis.
[1][2] The index set
However, the sum on the right must contain at most countably many non-zero terms, to have meaning.
[3] This definition is independent of the choice of the orthonormal basis.
In finite-dimensional Euclidean space, the Hilbert–Schmidt norm
is identical to the Frobenius norm.
The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis.
As for any bounded operator,
in the first formula, obtain
An important class of examples is provided by Hilbert–Schmidt integral operators.
Every bounded operator with a finite-dimensional range (these are called operators of finite rank) is a Hilbert–Schmidt operator.
The identity operator on a Hilbert space is a Hilbert–Schmidt operator if and only if the Hilbert space is finite-dimensional.
, which is a continuous linear operator of rank 1 and thus a Hilbert–Schmidt operator; moreover, for any bounded linear operator
Tr
{\displaystyle \operatorname {Tr} \left(A\left(x\otimes y\right)\right)=\left\langle Ax,y\right\rangle }
is a bounded compact operator with eigenvalues
ℓ
, where each eigenvalue is repeated as often as its multiplicity, then
, in which case the Hilbert–Schmidt norm of
is a measure space, then the integral operator
[5] 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
{\displaystyle \langle A,B\rangle _{\text{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, denoted by BHS(H) or B2(H), 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 norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space).
[4] The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with the Hilbert–Schmidt norm).
[4] The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, H is finite-dimensional.