Wold decomposition

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In operator theory, the Wold decomposition, or Wold-von Neumann decomposition, is a classification theorem for isometric linear operators on a given Hilbert space. It states that any isometry is a direct sums 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.

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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 = (\oplus_{\alpha \in A} S) \oplus U

for some index set A, where S in the unilateral shift on a Hilbert space Hα, and U is an unitary operator (possible vacuous). The family {Hα} consists of isomorphic Hilbert spaces.

A proof can be sketched as follows. Successive applications of V gives 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. If one defines

M_i = H_i \ominus H_{i+1} \quad \mbox{for} \quad i \geq 1 \;,

then

H = (\oplus_{i \geq 1} M_i) \oplus (\cap_{i \geq 1} 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. an unitary operator U.

Furthermore, each Mi is isomorphic to another, with V being an isomorphicm 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.

[edit] 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 \le \alpha \le N} S .

In this terminology, the Wold decomposition expresses an isometry as a direct sum of a pure isometry and an unitary.

A subspace M is called a wandering subspace of V if Vn(M) ⊥ Vm(M) for all n ≠ m. In particular, each Mi defined above is a wandering subspace of V.

[edit] A sequence of isometries

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

[edit] 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) = {Tf + K | Tf is a Toeplitz operator with continuous symbol f ∈ C(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_fT_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 f → f(U), and define

\Phi : C^*(S) \rightarrow C^*(V) \quad \mbox{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 f ∈ C(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.

[edit] References

  • L. Coburn, The C*-algebra of an isometry, Bull. Amer. Math. Soc. 73, 1967, 722-726.
  • T. Constantinescu, Schur Parameters, Dilation and Factorization Problems, Birkhauser Verlag, Vol. 82, 1996.
  • R.G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, 1972.
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