Hilbert–Schmidt operator

(Redirected from Hilbert-Schmidt operators)

In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator that acts on a Hilbert space and has finite Hilbert–Schmidt norm

where is an orthonormal basis.[1][2] The index set need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning.[3] This definition is independent of the choice of the orthonormal basis. In finite-dimensional Euclidean space, the Hilbert–Schmidt norm is identical to the Frobenius norm.

‖·‖HS is well defined

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The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if   and   are such bases, then   If   then   As for any bounded operator,   Replacing   with   in the first formula, obtain   The independence follows.

Examples

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An important class of examples is provided by Hilbert–Schmidt integral operators. Every bounded operator with a finite-dimensional range (these are called operators of finite rank) is a Hilbert–Schmidt operator. The identity operator on a Hilbert space is a Hilbert–Schmidt operator if and only if the Hilbert space is finite-dimensional. Given any   and   in  , define   by  , which is a continuous linear operator of rank 1 and thus a Hilbert–Schmidt operator; moreover, for any bounded linear operator   on   (and into  ),  .[4]

If   is a bounded compact operator with eigenvalues   of  , where each eigenvalue is repeated as often as its multiplicity, then   is Hilbert–Schmidt if and only if  , in which case the Hilbert–Schmidt norm of   is  .[5]

If  , where   is a measure space, then the integral operator   with kernel   is a Hilbert–Schmidt operator and  .[5]

Space of Hilbert–Schmidt operators

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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

 

The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on H. They also form a Hilbert space, denoted by BHS(H) or B2(H), which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces

 

where H is the dual space of H. The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space).[4] The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with the Hilbert–Schmidt norm).[4]

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

Properties

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  • Every Hilbert–Schmidt operator T : HH is a compact operator.[5]
  • A bounded linear operator T : HH is Hilbert–Schmidt if and only if the same is true of the operator  , in which case the Hilbert–Schmidt norms of T and |T| are equal.[5]
  • Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact operators.[5]
  • If   and   are Hilbert–Schmidt operators between Hilbert spaces then the composition   is a nuclear operator.[3]
  • If T : HH is a bounded linear operator then we have  .[5]
  • T is a Hilbert–Schmidt operator if and only if the trace   of the nonnegative self-adjoint operator   is finite, in which case  .[1][2]
  • If T : HH is a bounded linear operator on H and S : HH is a Hilbert–Schmidt operator on H then  ,  , and  .[5] In particular, the composition of two Hilbert–Schmidt operators is again Hilbert–Schmidt (and even a trace class operator).[5]
  • The space of Hilbert–Schmidt operators on H is an ideal of the space of bounded operators   that contains the operators of finite-rank.[5]
  • If A is a Hilbert–Schmidt operator on H then   where   is an orthonormal basis of H, and   is the Schatten norm of   for p = 2. In Euclidean space,   is also called the Frobenius norm.

See also

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References

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  1. ^ a b Moslehian, M. S. "Hilbert–Schmidt Operator (From MathWorld)".
  2. ^ a b Voitsekhovskii, M. I. (2001) [1994], "Hilbert-Schmidt operator", Encyclopedia of Mathematics, EMS Press
  3. ^ a b Schaefer 1999, p. 177.
  4. ^ a b c Conway 1990, p. 268.
  5. ^ a b c d e f g h i Conway 1990, p. 267.