The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias.
The defect indices of T are the pair The defect operators and the defect indices are a measure of the non-unitarity of T. A contraction T on a Hilbert space can be canonically decomposed into an orthogonal direct sum where U is a unitary operator and Γ is completely non-unitary in the sense that it has no non-zero reducing subspaces on which its restriction is unitary.
The explicit description of contractions leads to (operator-)parametrizations of positive and unitary matrices.
The operator U is called a dilation of T and is uniquely determined if U is minimal, i.e. K is the smallest closed subspace invariant under U and U* containing H. In fact define[1] the orthogonal direct sum of countably many copies of H. Let V be the isometry on
[2] Let G be a group, U(g) a unitary representation of G on a Hilbert space K and P an orthogonal projection onto a closed subspace H = PK of K. The operator-valued function with values in operators on K satisfies the positive-definiteness condition where Moreover, Conversely, every operator-valued positive-definite function arises in this way.
Recall that every (continuous) scalar-valued positive-definite function on a topological group induces an inner product and group representation φ(g) = 〈Ug v, v〉 where Ug is a (strongly continuous) unitary representation (see Bochner's theorem).
be the space of functions on G of finite support with values in H with inner product G acts unitarily on
The same method can be applied to prove a second dilation theorem of Sz._Nagy for a one-parameter strongly continuous contraction semigroup T(t) (t ≥ 0) on a Hilbert space H. Cooper (1947) had previously proved the result for one-parameter semigroups of isometries,[3] The theorem states that there is a larger Hilbert space K containing H and a unitary representation U(t) of R such that and the translates U(t)H generate K. In fact T(t) defines a continuous operator-valued positove-definite function Φ on R through for t > 0.
The previous construction yields a minimal unitary representation U(t) and projection P. The Hille–Yosida theorem assigns a closed unbounded operator A to every contractive one-parameter semigroup T'(t) through where the domain on A consists of all ξ for which this limit exists.
When A is a self-adjoint operator in the sense of the spectral theorem and this notation is used more generally in semigroup theory.
[4] Let T be totally non-unitary contraction on H. Then the minimal unitary dilation U of T on K ⊃ H is unitarily equivalent to a direct sum of copies the bilateral shift operator, i.e. multiplication by z on L2(S1).
[5] If P is the orthogonal projection onto H then for f in L∞ = L∞(S1) it follows that the operator f(T) can be defined by Let H∞ be the space of bounded holomorphic functions on the unit disk D. Any such function has boundary values in L∞ and is uniquely determined by these, so that there is an embedding H∞ ⊂ L∞.
In that case fr(T) is defined by the holomorphic functional calculus and f (T ) can be defined by The map sending f to f(T) defines an algebra homomorphism of H∞ into bounded operators on H. Moreover, if then This map has the following continuity property: if a uniformly bounded sequence fn tends almost everywhere to f, then fn(T) tends to f(T) in the strong operator topology.
It has the form φ ⋅ H∞ where g is an inner function, i.e. such that |φ| = 1 on S1: φ is uniquely determined up to multiplication by a complex number of modulus 1 and is called the minimal function of T. It has properties analogous to the minimal polynomial of a matrix.
The minimal function φ admits a canonical factorization where |c|=1, B(z) is a Blaschke product with and P(z) is holomorphic with non-negative real part in D. By the Herglotz representation theorem, for some non-negative finite measure μ on the circle: in this case, if non-zero, μ must be singular with respect to Lebesgue measure.
φ reduces to a Blaschke product exactly when H equals the closure of the direct sum (not necessarily orthogonal) of the generalized eigenspaces[6] Two contractions T1 and T2 are said to be quasi-similar when there are bounded operators A, B with trivial kernel and dense range such that The following properties of a contraction T are preserved under quasi-similarity: Two quasi-similar C0 contractions have the same minimal function and hence the same spectrum.
A C0 contraction operator T is multiplicity free if and only if it is quasi-similar to a Jordan block (necessarily corresponding the one corresponding to its minimal function).
with the λi's distinct, of modulus less than 1, such that and (ei) is an orthonormal basis, then S, and hence T, is C0 and multiplicity free.