# 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 every 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(H) be the bounded operators on H, and V ∈ L(H) be an isometry. The Wold decomposition states that every isometry V takes the form
    $V=\left(\bigoplus _{\alpha \in A}S\right)\oplus U$
for some index set A, where S is the unilateral shift on a Hilbert space H$_α$, and U is a unitary operator (possible vacuous). The family {H$_α$} 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:
    $H=H\supset V(H)\supset V^{2}(H)\supset \cdots =H_{0}\supset H_{1}\supset H_{2}\supset \cdots ,$
where V(H) denotes the range of V. The above defined H$_i$ = V$^i$(H). If one defines
    $M_{i}=H_{i}\ominus H_{i+1}=V^{i}(H\ominus V(H))\quad {\text{for}}\quad i\geq 0\;,$
then
    $H=\left(\bigoplus _{i\geq 0}M_{i}\right)\oplus \left(\bigcap _{i\geq 0}H_{i}\right)=K_{1}\oplus K_{2}.$
It is clear that K$_1$ and K$_2$ are invariant subspaces of V.
So V(K$_2$) = K$_2$. In other words, V restricted to K$_2$ is a surjective isometry, i.e., a unitary operator U.
Furthermore, each M$_i$ is isomorphic to another, with V being an isomorphism between M$_i$ and M$_{i+1}$: V "shifts" M$_i$ to M$_{i+1}$. Suppose the dimension of each M$_i$ is some cardinal number α. We see that K$_1$ can be written as a direct sum Hilbert spaces
    $K_{1}=\oplus H_{\alpha }$
where each H$_α$ is an invariant subspaces of V and V restricted to each H$_α$ is the unilateral shift S. Therefore
    $V=V\vert _{K_{1}}\oplus V\vert _{K_{2}}=\left(\bigoplus _{\alpha \in A}S\right)\oplus U,$
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, ⋂$_{i≥0}$H$_i$={0}.{\textstyle \bigcap _{i\geq 0}H_{i}=\{0\}.} 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
    $V=\bigoplus _{1\leq \alpha \leq N}S.$
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 V$^n$(M) ⊥ V$^m$(M) for all n ≠ m. In particular, each M$_i$ 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(H). Denote by C*(V) the C*-algebra generated by V, i.e. C*(V) is the norm closure of polynomials in V and V*. The Wold decomposition can be applied to characterize C*(V).
Let C(T) be the continuous functions on the unit circle T. We recall that the C*-algebra C*(S) generated by the unilateral shift S takes the following form
    C*(S) = {T$_f$ + K | T$_f$ is a Toeplitz operator with continuous symbol f ∈ C(T) and K is a compact operator}.
In this identification, S = T$_z$ where z is the identity function in C(T). The algebra C*(S) is called the Toeplitz algebra.
Theorem (Coburn) C*(V) is isomorphic to the Toeplitz algebra and V is the isomorphic image of T$_z$.
The proof hinges on the connections with C(T), 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:
* $T_{f}+T_{g}=T_{f+g}.\,$
* $T_{f}^{*}=T_{\bar {f}}.$
* The semicommutator $T_{f}T_{g}-T_{fg}\,$ is compact.
The Wold decomposition says that V is the direct sum of copies of T$_z$ and then some unitary U:
    $V=\left(\bigoplus _{\alpha \in A}T_{z}\right)\oplus U.$
So we invoke the continuous functional calculus f → f(U), and define
    $\Phi :C^{*}(S)\rightarrow C^{*}(V)\quad {\text{by}}\quad \Phi (T_{f}+K)=\bigoplus _{\alpha \in A}(T_{f}+K)\oplus f(U).$
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 T$_f$ is not compact for any non-zero f ∈ C(T) and thus T$_f$ + K = 0 implies f = 0. Since the range of Φ is a C*-algebra, Φ is surjective by the minimality of C*(V). 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.