# Wold decomposition

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In operator theory, the **Wold decomposition**, named after Herman Wold, or **Wold-von Neumann decomposition**, after Wold and John von Neumann, 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.

## 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

for some index set *A*, where *S* in the unilateral shift on a Hilbert space *H _{α}*, and

*U*is an unitary operator (possible vacuous). The family {

*H*} consists of isomorphic Hilbert spaces.

_{α}A proof can be sketched as follows. Successive applications of *V* give a descending sequences of copies of *H* isomorphically embedded in itself:

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

then

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. an unitary operator *U*.

Furthermore, each *M _{i}* is isomorphic to another, with

*V*being an isomorphism between

*M*and

_{i}*M*

_{i+1}:

*V*"shifts"

*M*to

_{i}*M*

_{i+1}. Suppose the dimension of each

*M*is some cardinal number

_{i}*α*. We see that

*K*

_{1}can be written as a direct sum Hilbert spaces

where each *H _{α}* is an invariant subspaces of

*V*and

*V*restricted to each

*H*is the unilateral shift

_{α}*S*. Therefore

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}. 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

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 *V ^{n}*(

*M*) ⊥

*V*(

^{m}*M*) for all

*n ≠ m*. In particular, each

*M*defined above is a wandering subspace of

_{i}*V*.

## A sequence of isometries

Template:Expand section 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*is a Toeplitz operator with continuous symbol_{f}*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 an unitary operator is contained in the circle **T**.

The following properties of the Toeplitz algebra will be needed:

The Wold decomposition says that *V* is the direct sum of copies of *T _{z}* and then some unitary

*U*:

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

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,
*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.

- Marvin Rosenblum and James Rovnyak,
*Hardy Classes and Operator Theory*, Oxford University Press, 1985.