# Hilbert-Schmidt operator

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An operator acting on a Hilbert space such that for any orthonormal basis in the following condition is met:

(however, this need be true for some basis only). A Hilbert–Schmidt operator is a compact operator for which the condition

applies to its -numbers and its eigen values ; here is a trace-class operator ( is the adjoint of and is the trace of an operator ). The set of all Hilbert–Schmidt operators on a fixed space forms a Hilbert space with scalar product

If is the resolvent of and

is its regularized characteristic determinant, then the Carleman inequality

holds.

A typical representative of a Hilbert–Schmidt operator is a Hilbert–Schmidt integral operator (which explains the origin of the name).