Hilbert–Schmidt operator

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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 is the norm of H, an orthonormal basis of H.[1][2] Note that the index set need not be countable; however, at most countably many terms will be non-zero.[3] These definitions are independent of the choice of the basis. In finite-dimensional Euclidean space, the Hilbert–Schmidt norm is identical to the Frobenius norm.

Definition[edit]

Suppose that is a Hilbert space. If is an orthonormal basis of H then for any linear operator A on H define:

where this sum may be finite or infinite. Note that this value is actually independent of the orthonormal basis of H that is chosen. Moreover, if the Hilbert–Schmidt norm is finite then the convergence of the sum necessitates that at most countably many of the terms are non-zero (even if I is uncountable). If A is a bounded linear operator then we have .[4]

A bounded operator A on a Hilbert space is a Hilbert–Schmidt operator if is finite. Equivalently, A is a Hilbert–Schmidt operator if the trace of the nonnegative self-adjoint operator is finite, in which case .[1][2]

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.

Examples[edit]

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 ), .[5]

If is a bounded compact operator with eigenvalues , 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 .[4]

If , where is a measure space, then the integral operator with kernel is a Hilbert–Schmidt operator and .[4]

Space of Hilbert–Schmidt operators[edit]

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).[5] 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).[5]

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

Properties[edit]

  • Every Hilbert–Schmidt operator T : HH is a compact operator.[4]
  • 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.[4]
  • Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact operators.[4]
  • 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 .[4]
  • If T : HH is a bounded linear operator on H and S : HH is a Hilbert–Schmidt operator on H then , , and .[4] In particular, the composition of two Hilbert–Schmidt operators is again Hilbert–Schmidt (and even a trace class operator).[4]
  • The space of Hilbert–Schmidt operators on H is an ideal of the space of bounded operators that contains the operators of finite-rank.[4]

See also[edit]

References[edit]

  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 d e f g h i j Conway 1990, p. 267.
  5. ^ a b c Conway 1990, p. 268.