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 A* is the adjoint of A. Note that the index set need not be countable; however, at most countably many terms will be non-zero. These definitions are independent of the choice of the basis.
In finite-dimensional Euclidean space, the Hilbert–Schmidt norm is identical to the Frobenius norm.

Definition

Suppose that is a Hilbert space.
If is an orthonormal basis of H then for any linear operator A on H define:
where this sum may be finite or infinite.
Note that this value actually independent of the orthonormal basis of H that is chosen.
Moreover, if the Hilbert-Schmidt norm is finite then the convergence of the sum converges necessitates that at most countably many of the terms are non-zero.
If A is a bounded linear operator then we have.
A bounded operator A on a Hilbert space is a Hilbert-Schmidt operator if is finite.
Equivalently, A is a Hilbert-Schmidt operator if the trace of the nonnegative self-adjoint operator is finite, in which case.
If A is an Hilbert–Schmidt operator on H then
where an orthonormal basis of H,, and is the Schatten norm of for.
In Euclidean space, is also called Frobenius norm.

Examples

An important class of examples is provided by Hilbert–Schmidt integral operators.
Every bounded operator with a finite-dimensional range 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.
Given any x and y in H, define by = <z, y> x, which is a continuous linear operator of rank 1 and thus a Hilbert-Schmidt operator;
moreover, for any bounded linear operator A on H,.
If T : H → H is a bounded compact operator with eigenvalues l1, l2,... where each eigenvalue is repeated as often as its multiplicity, then T is Hilbert-Schmidt if and only if, in which case the Hilbert-Schmidt norm of T is.
If, where is a measure space, then the integral operator with kernel k is a Hilbert-Schmidt operator and.

Space of Hilbert–Schmidt operators

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, denoted by BHS or B2, 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.
The space of all bounded linear operators of finite rank is a dense subset of the space of Hilbert–Schmidt operators.
The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, H is finite-dimensional.

Properties