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Hilbert–Schmidt operator

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. 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. 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.

‖·‖ is well defined

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

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 ), .

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 .

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

Space of Hilbert–Schmidt operators

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 . They also form a Hilbert space, denoted by or , which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces

where is the dual space of . 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). 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).

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

Properties

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

See also

References