Hilbert–Schmidt operator

In mathematics, a Hilbert–Schmidt operator, named for David Hilbert and Erhard Schmidt, is a bounded operator A on a Hilbert space H with finite Hilbert–Schmidt norm

where ${\displaystyle \|\ \|}$ is the norm of H, ${\displaystyle \{e_{i}:i\in I\}}$ an orthonormal basis of H, and Tr is the trace of a nonnegative self-adjoint operator. Note that the index set need not be countable. This definition is independent of the choice of the basis, and therefore

for ${\displaystyle A_{i,j}=\langle e_{i},Ae_{j}\rangle }$ and ${\displaystyle \|A\|_{2}}$ the Schatten norm of ${\displaystyle A}$ for p = 2. In Euclidean space ${\displaystyle \|\ \|_{HS}}$ is also called Frobenius norm, named for Ferdinand Georg Frobenius.

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