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

\|A\|^2_{HS}={\rm Tr} |(A^{{}^*}A)|:= \sum_{i \in I} \|Ae_i\|^2

where \|\ \| is the norm of H and \{e_i : i\in I\} an orthonormal basis of H for an index set I.[1][2] Note that the index set need not be countable. This definition is independent of the choice of the basis, and therefore

\|A\|^2_{HS}=\sum_{i,j} |A_{i,j}|^2 = \|A\|^2_2

for A_{i,j}=\langle e_i, Ae_j \rangle and \|A\|_2 the Schatten norm of A. In Euclidean space \|\ \|_{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

\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

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.

Functionals[edit]

A mapping \phi:H_1\times H_2\to C is a Hilbert-Schmidt functional if it is a bounded bilinear functional.[3]

A bounded linear mapping L:H_1\times H_2\to K is weakly Hilbert-Schmidt if for all v\in K the mapping

\phi_v = (u_1,u_2)\mapsto\langle L(u_1,u_2), v\rangle

is a Hilbert-Schmidt functional and \|\phi_v\|\leq M\|v\| for some real number M\geq 0.[4]

References[edit]

  1. ^ Moslehian, M.S. "Hilbert–Schmidt Operator (From MathWorld)". 
  2. ^ Voitsekhovskii, M.I. (2001), "H/h047350", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
  3. ^ Kadison, Richard V.; Ringrose, John R. (1997), Fundamentals of the theory of operator algebras. Vol. I, Graduate Studies in Mathematics 15, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0819-1, MR 1468229  (see p. 127)
  4. ^ Kadison and Ringrose, (see p. 131)