Direct integral

People seem to really like replacing sums by integrals. It turns out that even something like a direct sum of Hilbert spaces can be generalized. The decomposition of the regular representation of the circle \mathbb{T} (which is a Lie group) into irreducible unitary representations gives us the theory of Fourier series. One thing that confused me for a while is another thing called the “Fourier transform”, which is when you write a function on \mathbb{R} (rather than a function on the circle/periodic function on \mathbb{R}) as an integral rather than a series. However, I realized it’s completely natural to do so: the analogous decomposition for the additive group of real numbers (which is also a Lie group) gives something a bit different: you can’t decompose L^2(\mathbb{R}) as an orthogonal direct sum like you did to L^2(\mathbb{T}), but you can decompose it into something called a “direct integral”…

Suppose you have a set (which will act as an “index set” for your spaces) X equipped with a non-negative measure, \mu, and for each x \in X you have an associated Hilbert space \mathfrak{H}_x. Suppose each \mathfrak{H}_x is of the same dimension, so we can identify them all with one space \mathfrak{H}. We denote by \mathcal{H} the space of functions \xi : X \to \mathfrak{H} such that for any v \in \mathfrak{H}, the function x \mapsto (\xi(x), v) is \mu-measurable, and the function x \mapsto \| \xi(x) \| is square-integrable w.r.t. \mu, that is,

\displaystyle \int_X \| \xi(x) \|^2 \; d\mu(x) < \infty.

Then \mathcal{H} becomes (as one can show) a Hilbert space under the obvious pointwise operations of vector addition and scalar multiplication, and the scalar product

\displaystyle (\xi, \eta) = \int_X (\xi(x), \eta(x)) \; d \mu(x).

\mathcal{H} is called the continuous direct sum (or direct integral) of the spaces \mathfrak{H}_x with respect to \mu, and we write it as

\displaystyle \mathcal{H} =: \int_X^{\bigoplus} \mathfrak{H}_x \; d\mu(x).


About mlbaker

just another guy trying to make the diagrams commute.
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One Response to Direct integral

  1. Bernd says:

    And that is where the fun begins: F.e. every unitary representation of a locally compact group on a Hilbert space is a direct integral of irreducible representations. The simplest example is probably the fact that by using the Fourier transform one has that the representation of \mathbb{R} on L^2(\mathbb R) is nothing else but the direct integral of copies of \mathbb{R}, or more precisely of the Hilbert spaces H_y=  e^{icy} on which \mathbb{R} clearly acts.

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