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Concept
Hilbert–Schmidt operator
Summary
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
where is an orthonormal basis. The index set 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, the Hilbert–Schmidt norm is identical to the Frobenius norm.
The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if and are such bases, then
If then As for any bounded operator, Replacing with in the first formula, obtain The independence follows.
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.
Given any and in , define by , which is a continuous linear operator of rank 1 and thus a Hilbert–Schmidt operator;
moreover, for any bounded linear operator on (and into ), .
If is a bounded compact operator with eigenvalues of , where each eigenvalue is repeated as often as its multiplicity, then is Hilbert–Schmidt if and only if , in which case the Hilbert–Schmidt norm of is .
If , where is a measure space, then the integral operator with kernel is a Hilbert–Schmidt operator and .
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(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.
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