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(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

${\displaystyle V=\left(\bigoplus _{\alpha \in A}S\right)\oplus U}$

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:

${\displaystyle H=H\supset V(H)\supset V^{2}(H)\supset \cdots =H_{0}\supset H_{1}\supset H_{2}\supset \cdots ,}$

where V(H) denotes the range of V. The above defined H_i = Vⁱ(H). If one defines

${\displaystyle M_{i}=H_{i}\ominus H_{i+1}=V^{i}(H\ominus V(H))\quad {\text{for}}\quad i\geq 0\;,}$

then

${\displaystyle H=\left(\bigoplus _{i\geq 0}M_{i}\right)\oplus \left(\bigcap _{i\geq 0}H_{i}\right)=K_{1}\oplus K_{2}.}$

It is clear that 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 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 K₁ can be written as a direct sum Hilbert spaces

${\displaystyle K_{1}=\oplus H_{\alpha }}$

where each H_α is an invariant subspaces of V and V restricted to each H_α is the unilateral shift S. Therefore

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

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, ${\textstyle \bigcap _{i\geq 0}H_{i}=\{0\}.}$ 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

${\displaystyle V=\bigoplus _{1\leq \alpha \leq N}S.}$

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 Vⁿ(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:

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

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

${\displaystyle V=\left(\bigoplus _{\alpha \in A}T_{z}\right)\oplus U.}$

So we invoke the continuous functional calculus f → f(U), and define

${\displaystyle \Phi :C^{*}(S)\rightarrow C^{*}(V)\quad {\text{by}}\quad \Phi (T_{f}+K)=\bigoplus _{\alpha \in A}(T_{f}+K)\oplus f(U).}$

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.

References

• Coburn, L. (1967). "The C*-algebra of an isometry". Bull. Amer. Math. Soc. 73 (5): 722–726. doi:10.1090/S0002-9904-1967-11845-7.
• Constantinescu, T. (1996). Schur Parameters, Factorization and Dilation Problems. Operator Theory, Advances and Applications. Vol. 82. Birkhäuser. ISBN 3-7643-5285-X.
• Douglas, R. G. (1972). Banach Algebra Techniques in Operator Theory. Academic Press. ISBN 0-12-221350-5.
• Rosenblum, Marvin; Rovnyak, James (1985). Hardy Classes and Operator Theory. Oxford University Press. ISBN 0-19-503591-7.