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 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 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 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 isomorphism 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
where each Hα is an invariant subspaces of V and V restricted to each Hα is the unilateral shift S. Therefore
which is a Wold decomposition of V.
Read more about this topic: Wold Decomposition
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