Hilbert–Schmidt operator

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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

${\displaystyle \|A\|_{HS}^{2}={\rm {Tr}}(A^{*}A):=\sum _{i\in I}\|Ae_{i}\|^{2}}$

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.[1][2] Note that the index set need not be countable. This definition is independent of the choice of the basis, and therefore

${\displaystyle \|A\|_{HS}^{2}=\sum _{i,j}|A_{i,j}|^{2}=\|A\|_{2}^{2}}$

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.

The product of two Hilbert–Schmidt operators has finite trace class norm; therefore, if A and B are two Hilbert–Schmidt operators, the Hilbert–Schmidt inner product can be defined as

${\displaystyle \langle A,B\rangle _{\mathrm {HS} }=\operatorname {Tr} (A^{*}B)=\sum _{i}\langle Ae_{i},Be_{i}\rangle .}$

The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on H. They also form a Hilbert space, which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces

${\displaystyle H^{*}\otimes H,\,}$

where H is the dual space of H.

The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, H is finite-dimensional.

An important class of examples is provided by Hilbert–Schmidt integral operators.

Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact.

References

1. ^ Moslehian, M.S. "Hilbert–Schmidt Operator (From MathWorld)".
2. ^ Voitsekhovskii, M.I. (2001) [1994], "H/h047350", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4