In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain Ω in "n"-dimensional Euclidean space R"n", then the square-integrable function "k" : Ω × Ω → C belonging to "L"2(Ω×Ω) such that
formula_1
is called a Hilbert–Schmidt kernel and the associated integral operator "T" : "L"2(Ω) → "L"2(Ω) given by
formula_2
is called a Hilbert–Schmidt integral operator.
Then "T" is a Hilbert–Schmidt operator with Hilbert–Schmidt norm
formula_3
Hilbert–Schmidt integral operators are both continuous and compact.
The concept of a Hilbert–Schmidt operator may be extended to any locally compact Hausdorff spaces. Specifically, let "L"2("X") be a separable Hilbert space and "X" a locally compact Hausdorff space equipped with a positive Borel measure. The initial condition on the kernel "k" on Ω ⊆ Rn can be reinterpreted as demanding "k" belong to "L"2("X × X"). Then the operator
formula_4
is compact. If 
formula_5
then "T" is also self-adjoint and so the spectral theorem applies. This is one of the fundamental constructions of such operators, which often reduces problems about infinite-dimensional vector spaces to questions about well-understood finite-dimensional eigenspaces.
Notes.
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