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
Famous quotes containing the word details:
“Then he told the news media
the strange details of his death
and they hammered him up in the marketplace
and sold him and sold him and sold him.
My death the same.”
—Anne Sexton (1928–1974)
“Different persons growing up in the same language are like different bushes trimmed and trained to take the shape of identical elephants. The anatomical details of twigs and branches will fulfill the elephantine form differently from bush to bush, but the overall outward results are alike.”
—Willard Van Orman Quine (b. 1908)
“Working women today are trying to achieve in the work world what men have achieved all along—but men have always had the help of a woman at home who took care of all the other details of living! Today the working woman is also that woman at home, and without support services in the workplace and a respect for the work women do within and outside the home, the attempt to do both is taking its toll—on women, on men, and on our children.”
—Jeanne Elium (20th century)