Wold's decomposition

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

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

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:

:$H\; =\; H\; \backslash supset\; V(H)\; \backslash supset\; V^2\; (H)\; \backslash supset\; \backslash cdots\; =\; H\_0\; \backslash supset\; H\_1\; \backslash supset\; H\_2\; \backslash supset\; \backslash cdots,$

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

:$M\_i\; =\; H\_i\; \backslash ominus\; H\_\{i+1\}\; =\; V^i\; (H\; \backslash ominus\; V(H))\; \backslash quad\; \backslash text\{for\}\; \backslash quad\; i\; \backslash geq\; 0\; \backslash ;,$

then

:$H\; =\; \backslash left(\; \backslash bigoplus\_\{i\; \backslash geq\; 0\}\; M\_i\; \backslash right)\; \backslash oplus\; \backslash left(\; \backslash bigcap\_\{i\; \backslash geq\; 0\}\; H\_i\; \backslash right)\; =\; K\_1\; \backslash oplus\; K\_2.$

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

:$K\_1\; =\; \backslash oplus\; H\_\{\backslash alpha\}$

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

:$V\; =\; V\; \backslash vert\_\{K\_1\}\; \backslash oplus\; V\backslash vert\_\{K\_2\}\; =\; \backslash left(\backslash bigoplus\_\{\backslash alpha\; \backslash in\; A\}\; S\; \backslash right)\; \backslash oplus\; U,$

which is a Wold decomposition of V.

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, $\backslash bigcap\_\{i\backslash ge0\}\; H\_i\; =\; \backslash \{0\backslash \}.$ 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

:$V\; =\; \backslash bigoplus\_\{1\; \backslash le\; \backslash alpha\; \backslash le\; N\}\; S\; .$

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.

{{Expand section|date=June 2008}}

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

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. $T\_f\; +\; T\_g\; =\; T\_\{f+g\}.\backslash ,$
2. $T\_f\; ^*\; =\; T\_\{\{\backslash bar\; f\}\}\; .$
3. The semicommutator $T\_fT\_g\; -\; T\_\{fg\}\; \backslash ,$ is compact.

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

:$V\; =\; \backslash left(\; \backslash bigoplus\_\{\backslash alpha\; \backslash in\; A\}\; T\_z\; \backslash right)\; \backslash oplus\; U.$

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

:$$

\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.

• {{cite journal |first=L. |last=Coburn |title=The C*-algebra of an isometry |journal=Bull. Amer. Math. Soc. |volume=73 |issue=5 |year=1967 |pages=722–726 |doi=10.1090/S0002-9904-1967-11845-7 |doi-access=free }}
• {{cite book |first=T. |last=Constantinescu |title=Schur Parameters, Factorization and Dilation Problems |publisher=Birkhäuser |series=Operator Theory, Advances and Applications |volume=82 |year=1996 |isbn=3-7643-5285-X |url=https://books.google.com/books?id=aEwoy7k8ufcC }}
• {{cite book |first=R. G. |last=Douglas |title=Banach Algebra Techniques in Operator Theory |publisher=Academic Press |year=1972 |isbn=0-12-221350-5 |url=https://books.google.com/books?id=_xvDk-mOiHIC }}
• {{cite book |first=Marvin |last=Rosenblum |first2=James |last2=Rovnyak |title=Hardy Classes and Operator Theory |publisher=Oxford University Press |year=1985 |isbn=0-19-503591-7 |url=https://books.google.com/books?id=PchPAQAAIAAJ }}

{{Functional analysis}}

{{Hilbert space}}

Category:Operator theory

Category:Invariant subspaces

Category:C*-algebras

Category:Theorems in functional analysis

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