Wold's decomposition


In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator.
In time series analysis, the theorem implies that any stationary discrete-time stochastic process can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a moving average process.

Details

Let H be a Hilbert space, L be the bounded operators on H, and V ∈ L be an isometry. The Wold decomposition states that every isometry V takes the form
for some index set A, where S is the unilateral shift on a Hilbert space Hα, and U is a unitary operator. The family consists of isomorphic Hilbert spaces.
A proof can be sketched as follows. Successive applications of V give a descending sequences of copies of H isomorphically embedded in itself:
where V denotes the range of V. The above defined Hi = Vi. If one defines
then
It is clear that K1 and K2 are invariant subspaces of V.
So V = K2. In other words, V restricted to K2 is a surjective isometry, i.e., a unitary operator U.
Furthermore, each Mi is isomorphic to another, with V being an isomorphism between Mi and Mi+1: V "shifts" Mi to Mi+1. Suppose the dimension of each Mi is some cardinal number α. We see that K1 can be written as a direct sum Hilbert spaces
where each Hα is an invariant subspaces of V and V restricted to each Hα is the unilateral shift S. Therefore
which is a Wold decomposition of V.

Remarks

It is immediate from the Wold decomposition that the spectrum of any proper, i.e. non-unitary, isometry is the unit disk in the complex plane.
An isometry V is said to be pure if, in the notation of the above proof, ∩i≥0 Hi =. The multiplicity of a pure isometry V is the dimension of the kernel of V*, i.e. the cardinality of the index set A in the Wold decomposition of V. In other words, a pure isometry of multiplicity N takes the form
In this terminology, the Wold decomposition expresses an isometry as a direct sum of a pure isometry and a unitary operator.
A subspace M is called a wandering subspace of V if Vn ⊥ Vm for all n ≠ m. In particular, each Mi defined above is a wandering subspace of V.

A sequence of isometries

The decomposition above can be generalized slightly to a sequence of isometries, indexed by the integers.

The C*-algebra generated by an isometry

Consider an isometry V ∈ L. Denote by C* the C*-algebra generated by V, i.e. C* is the norm closure of polynomials in V and V*. The Wold decomposition can be applied to characterize C*.
Let C be the continuous functions on the unit circle T. We recall that the C*-algebra C* generated by the unilateral shift S takes the following form
In this identification, S = Tz where z is the identity function in C. The algebra C* is called the Toeplitz algebra.
Theorem C* is isomorphic to the Toeplitz algebra and V is the isomorphic image of Tz.
The proof hinges on the connections with C, in the description of the Toeplitz algebra and that the spectrum of a unitary operator is contained in the circle T.
The following properties of the Toeplitz algebra will be needed:
  1. The semicommutator is compact.
The Wold decomposition says that V is the direct sum of copies of Tz and then some unitary U:
So we invoke the continuous functional calculus f → f, and define
One can now verify Φ is an isomorphism that maps the unilateral shift to V:
By property 1 above, Φ is linear. The map Φ is injective because Tf is not compact for any non-zero f ∈ C and thus Tf + K = 0 implies f = 0. Since the range of Φ is a C*-algebra, Φ is surjective by the minimality of C*. Property 2 and the continuous functional calculus ensure that Φ preserves the *-operation. Finally, the semicommutator property shows that Φ is multiplicative. Therefore the theorem holds.