direct integral of Hilbert spaces

Let X be a measure space with measure μ. To each point xX, assign a Hilbert spaceMathworldPlanetmath H(x). Then the direct integral of this family of Hilbert spaces indexed by the points of a measure space, denoted


is the set of all functions v:XxXH(x) (as usual in functional analysisMathworldPlanetmathPlanetmath, we regard two functions that disagree on a set of measure zero as the same) such that

  • v(x)H(x)

  • v(x)L2(X,μ)

Vector addition and scalar multiplication are defined pointwise: (u+v)(x)=u(x)+v(x) and (sv)(x)=sv(x). The inner productMathworldPlanetmath is defined as

u,v=Xu(x),v(x)𝑑x  .

A nice illustration of the direct integral is the expression of L2(2) as a direct integral of a continuousMathworldPlanetmath infinity of copies of L2(). Let the space H(k) consists of all functions of the form (x,y)2eikxg(y) such that -+|g(y)|2<. Define the inner product of two functions eikxg1(y)H(k) and eikxg2(y)H(k) as -+g1(y)g2(y). (By the way, note that H(k) is not a subset of L2(2) because the functions contained in H(k) are not square-integrable in the x variable.)

By the definition, the space -+H(k)𝑑k consists of functions (k,x,y)eikxg(k,y) such that


Elements of this space may be regarded as functions of x and y in a natural way:


By Parseval’s identityPlanetmathPlanetmath, g~L2(2).

The reason for choosing this example is that it illustrates two situations in which direct integrals are usually encountered. First, direct integrals can be used to provide an analogue of the decomposition of a vector spaceMathworldPlanetmath as a direct sumPlanetmathPlanetmath of eigenspacesMathworldPlanetmath of a matrix for operators with continuous spectra. The space H(k) in the example can be regarded as the “eigenspace” of the formally self-adjoint operator -i/x with eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath k. (It is not eigenspaces in the strict sense of the term since its elements are not square integrable in x.) The spectrum of this operator is the whole real axis and the direct integral of these “eigenspaces” is the whole vector space on which the operator acts.

A second situation which can be illustrated with this example is the use of direct integrals in group representationMathworldPlanetmathPlanetmath theory. Consider the group of translationsMathworldPlanetmathPlanetmath along the x axis. Each of the spaces H(k) transforms under a different representation of this group — under translation by an amount x0 along the x axis, an element of H(k) is multiplied by a factor eikx0. The direct integral is the infinite-dimensional analogue of the decomposition of a finite-dimensional vector space on which a group acts as a direct sum of irreducible representations.

Title direct integral of Hilbert spaces
Canonical name DirectIntegralOfHilbertSpaces
Date of creation 2013-03-22 14:43:57
Last modified on 2013-03-22 14:43:57
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 13
Author rspuzio (6075)
Entry type Definition
Classification msc 46C05
Defines direct integral