Wold's decomposition

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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(H) be the bounded operators on H, and VL(H) be an isometry. The Wold decomposition states that every isometry V takes the form

$V=(\oplus _{\alpha \in A}S)\oplus U$ for some index set A, where S is the unilateral shift on a Hilbert space Hα, and U is a unitary operator (possible vacuous). The family {Hα} 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 Hi = Vi(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

$H=(\oplus _{i\geq 0}M_{i})\oplus (\cap _{i\geq 0}H_{i})=K_{1}\oplus K_{2}.$ It is clear that K1 and K2 are invariant subspaces of V.

So V(K2) = 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

$K_{1}=\oplus H_{\alpha }$ where each Hα is an invariant subspaces of V and V restricted to each Hα is the unilateral shift S. Therefore

$V=V\vert _{K_{1}}\oplus V\vert _{K_{2}}=(\oplus _{\alpha \in A}S)\oplus U,$ 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 Hi = {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

$V=\oplus _{1\leq \alpha \leq N}S.$ 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 Vn(M) ⊥ Vm(M) for all nm. 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 VL(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) = {Tf + K | Tf is a Toeplitz operator with continuous symbol fC(T) and K is a compact operator}.

In this identification, S = Tz 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 Tz.

The proof hinges on the connections with C(T), in the description of the Toeplitz algebra and that the spectrum of an unitary operator is contained in the circle T.

The following properties of the Toeplitz algebra will be needed:

1. $T_{f}+T_{g}=T_{f+g}.\,$ 2. $T_{f}^{*}=T_{\bar {f}}.$ 3. The semicommutator $T_{f}T_{g}-T_{fg}\,$ is compact.

The Wold decomposition says that V is the direct sum of copies of Tz and then some unitary U:

$V=(\oplus _{\alpha \in A}T_{z})\oplus U.$ So we invoke the continuous functional calculus ff(U), and define

$\Phi :C^{*}(S)\rightarrow C^{*}(V)\quad {\text{by}}\quad \Phi (T_{f}+K)=\oplus _{\alpha \in A}(T_{f}+K)\oplus f(U).$ 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 Tf is not compact for any non-zero fC(T) 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*(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.