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

is an orthonormal basis. need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This definition is independent of the choice of the orthonormal basis. In finite-dimensional Euclidean space is identical to the Frobenius norm.

‖·‖HS is well defined

The independence follows.

Examples

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, which is a continuous linear operator of rank 1 and thus a Hilbert–Schmidt operator; moreover, for any bounded linear operator .

.

. is Hilbert-Schmidt then it may be written as an integral operator of this form.

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 . They also form a Hilbert space, denoted by or , which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces

where is the dual space of . 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). 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).

The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, is finite-dimensional.

Properties

• Every Hilbert–Schmidt operator is a compact operator.

• A bounded linear operator are equal.

• Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact operators.

• is a nuclear operator.

• If .

• is a Hilbert–Schmidt operator if and only if the trace.

• If is a bounded linear operator on and is a Hilbert–Schmidt operator on . In particular, the composition of two Hilbert–Schmidt operators is again Hilbert–Schmidt (and even a trace class operator).

• The space of Hilbert–Schmidt operators on is an ideal that contains the operators of finite-rank.

• If is a Hilbert–Schmidt operator on is an orthonormal basis of H is the Schatten norm for . In Euclidean space is also called the Frobenius norm.

See also

References