A Hilbert–Schmidt operator is a special type of compact linear operator acting on a Hilbert space, which can be characterized by certain properties of its kernel. Specifically, it is defined in the context of an inner product space, typically \(L^2\) spaces.
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A bounded linear operator on a Hilbert space is Hilbert-Schmidt when for one, equivalently every, orthonormal basis. Every Hilbert-Schmidt operator is a compact operator.