direct integral of Hilbert spaces
is the set of all functions (as usual in functional analysis, we regard two functions that disagree on a set of measure zero as the same) such that
A nice illustration of the direct integral is the expression of as a direct integral of a continuous infinity of copies of . Let the space consists of all functions of the form such that . Define the inner product of two functions and as . (By the way, note that is not a subset of because the functions contained in are not square-integrable in the variable.)
By the definition, the space consists of functions such that
Elements of this space may be regarded as functions of and in a natural way:
By Parseval’s identity, .
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 space as a direct sum of eigenspaces of a matrix for operators with continuous spectra. The space in the example can be regarded as the “eigenspace” of the formally self-adjoint operator with eigenvalue . (It is not eigenspaces in the strict sense of the term since its elements are not square integrable in .) 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 representation theory. Consider the group of translations along the axis. Each of the spaces transforms under a different representation of this group — under translation by an amount along the axis, an element of is multiplied by a factor . 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|
|Date of creation||2013-03-22 14:43:57|
|Last modified on||2013-03-22 14:43:57|
|Last modified by||rspuzio (6075)|