Wold's decomposition explained

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

V = \left( o plus _{\alpha} S\right) ⊕ 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 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 … = H _{0} \supset H _{1} \supset H _{2} \supset … ,

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

M _{i} = H _{i} \ominus H _{i+1} = V ^{i} (H \ominus V(H)) for i \geq 0 ,

then

H = \left( o plus _{i} M _{i} \right) ⊕ \left( c ap _{i} H _{i} \right) = K _{1} ⊕ 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

K _{1} = ⊕ 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}} ⊕ V\vert _{K _{2}} = \left( o plus _{\alpha} S \right) ⊕ 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, $\backslash bigcap\_\; H\_i\; =\; \backslash .$ 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 = o plus _{1} 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) = .

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} } .

1. The semicommutator

T _{fT} _{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( o plus _{\alpha} T _{z} \right) ⊕ U.

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

\Phi : C ^{*(S)} → C ^{*(V)} by \Phi(T _{f} + K) = o plus _{\alpha} (T _{f} + K) ⊕ 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

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