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.
For , the Cauchy-Schwarz inequality gives . The kernel norm is the Hilbert-Schmidt norm, which bounds the operator norm. Uniform kernel bounds apply also to spatially parametrized operators after integrating over position.
The Hilbert-Schmidt norm of a Hilbert-Schmidt operator is the square root of the displayed sum. The value is independent of the orthonormal basis, by the Parseval identity. For a finite matrix, it is the Frobenius norm.
The Frobenius norm is the finite-matrix Hilbert-Schmidt norm. It equals , is unchanged by unitary multiplication on either side, and is the square root of the sum of squared singular values. For unit vectors , , a useful sign-invariant measure of distance between rank-one orthogonal projection matrices.
For Hilbert-Schmidt operators and , the Hilbert-Schmidt inner product is . Its induced norm is the Hilbert-Schmidt norm.
If is a density operator with decreasing eigenvalues and has matrix rank at most , let project onto the image of . The Cauchy-Schwarz inequality for the Hilbert-Schmidt inner product givesThe Hermitian effect variational principle bounds by , proving the claimed estimate without assuming is invertible.
The Hilbert-Schmidt distance is .
A bounded operator on a separable Hilbert space is trace class when is Hilbert--Schmidt. Its trace norm is
An operator is trace class exactly when it factors as with and Hilbert--Schmidt. The factors can be chosen so that by using the polar decomposition of a bounded operator.
The Schatten norm Hölder inequality gives when . In particular, the product of two Hilbert-Schmidt operators is trace class and
For a trace-class operator, the series converges absolutely and is independent of the orthonormal basis. It satisfies .
Every bounded operator defines a functional on the trace-class operators, andFinite-rank density in makes every continuous functional arise uniquely this way.
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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.