# 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.

## Contents

## Details[edit]

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* is the unilateral shift on a Hilbert space *H _{α}*, and

*U*is a 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 *H*_{i} = *V*^{i}(*H*). 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., a 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[edit]

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 a unitary operator.

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*_{i} defined above is a wandering subspace of *V*.

## A sequence of isometries[edit]

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

## The C*-algebra generated by an isometry[edit]

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 an unitary operator is contained in the circle **T**.

The following properties of the Toeplitz algebra will be needed:

- The semicommutator is compact.

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[edit]

- 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.**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.