Contraction (operator theory)

Dilation theorem for contractions

Sz.-Nagy's dilation theorem, proved in 1953, states that for any contraction T on a Hilbert space H, there is a unitary operator U on a larger Hilbert space K ⊇ H such that if P is the orthogonal projection of K onto H then ${\displaystyle T^n=P{U}^n{P}^{\ast }}$ for all n > 0. 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

${\displaystyle {\displaystyle \mathcal{H}=H\oplus H\oplus H\oplus \cdots ,}}$

the orthogonal direct sum of countably many copies of H.

Let V be the isometry on ${\displaystyle \mathcal{H}}$ defined by

${\displaystyle {\displaystyle V({\xi }_1,{\xi }_2,{\xi }_3,\dots )=(T{\xi }_1,\sqrt{I-T^{\ast }T}{\xi }_1,{\xi }_2,{\xi }_3,\dots ).}}$

Let

${\displaystyle {\displaystyle \mathcal{K}=\mathcal{H}\oplus \mathcal{H}.}}$

Define a unitary W on ${\displaystyle \mathcal{K}}$ by

${\displaystyle {\displaystyle W(x,y)=(Vx+(I-V{V}^{\ast })y,-V^{\ast }y).}}$

W is then a unitary dilation of T with H considered as the first component of ${\displaystyle \mathcal{H}\subset \mathcal{K}}$.

The minimal dilation U is obtained by taking the restriction of W to the closed subspace generated by powers of W applied to H.

Dilation theorem for contraction semigroups

There is an alternative proof of Sz.-Nagy's dilation theorem, which allows significant generalization.

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

${\displaystyle {\displaystyle \mathrm{\Phi }(g)=PU(g)P,}}$

with values in operators on K satisfies the positive-definiteness condition

${\displaystyle \sum {\lambda }_i\overline{{\lambda }_j}\mathrm{\Phi }(g_j^{-1}{g}_i)=P{T}^{\ast }TP\ge 0,}$

where

${\displaystyle {\displaystyle T=\sum {\lambda }_{i}U(g_i).}}$

Moreover,

${\displaystyle {\displaystyle \mathrm{\Phi }(1)=P.}}$

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) = 〈U_g v, v〉 where U_g is a (strongly continuous) unitary representation (see Bochner's theorem). Replacing v, a rank-1 projection, by a general projection gives the operator-valued statement. In fact the construction is identical; this is sketched below.

Let ${\displaystyle \mathcal{H}}$ be the space of functions on G of finite support with values in H with inner product

${\displaystyle {\displaystyle (f_1,f_2)=\sum _{g,h}(\mathrm{\Phi }(h^{-1}g)f_1(g),f_2(h)).}}$

G acts unitarily on ${\displaystyle \mathcal{H}}$ by

${\displaystyle {\displaystyle U(g)f(x)=f(g^{-1}x).}}$

Moreover, H can be identified with a closed subspace of ${\displaystyle \mathcal{H}}$ using the isometric embedding sending v in H to f_v with

${\displaystyle f_v(g)={\delta }_{g,1}v.}$

If P is the projection of ${\displaystyle \mathcal{H}}$ onto H, then

${\displaystyle {\displaystyle PU(g)P=\mathrm{\Phi }(g),}}$

using the above identification.

When G is a separable topological group, Φ is continuous in the strong (or weak) operator topology if and only if U is.

In this case functions supported on a countable dense subgroup of G are dense in ${\displaystyle \mathcal{H}}$, so that ${\displaystyle \mathcal{H}}$ is separable.

When G = Z any contraction operator T defines such a function Φ through

${\displaystyle {\displaystyle \mathrm{\Phi }(0)=I,\mathrm{\Phi }(n)=T^n,\mathrm{\Phi }(-n)=(T^{\ast }{)}^n,}}$

for n > 0. The above construction then yields a minimal unitary dilation.

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.

The theorem states that there is a larger Hilbert space K containing H and a unitary representation U(t) of R such that

${\displaystyle {\displaystyle T(t)=PU(t)P}}$

and the translates U(t)H generate K.

In fact T(t) defines a continuous operator-valued positive-definite function Φ on R through

${\displaystyle {\displaystyle \mathrm{\Phi }(0)=I,\mathrm{\Phi }(t)=T(t),\mathrm{\Phi }(-t)=T(t{)}^{\ast },}}$

for t > 0. Φ is positive-definite on cyclic subgroups of R, by the argument for Z, and hence on R itself by continuity.

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

${\displaystyle {\displaystyle A\xi =\underset{t\downarrow 0}{lim}\frac{1}{t}(T(t)-I)\xi ,}}$

where the domain on A consists of all ξ for which this limit exists.

A is called the generator of the semigroup and satisfies

${\displaystyle {\displaystyle -\mathrm{\Re }(A\xi ,\xi )\ge 0}}$

on its domain. When A is a self-adjoint operator

${\displaystyle {\displaystyle T(t)=e^{At},}}$

in the sense of the spectral theorem and this notation is used more generally in semigroup theory.

The cogenerator of the semigroup is the contraction defined by

${\displaystyle {\displaystyle T=(A+I)(A-I{)}^{-1}.}}$

A can be recovered from T using the formula

${\displaystyle {\displaystyle A=(T+I)(T-I{)}^{-1}.}}$

In particular a dilation of T on K ⊃ H immediately gives a dilation of the semigroup.

Functional calculus

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).

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

${\displaystyle {\displaystyle f(T)\xi =Pf(U)\xi .}}$

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^∞.

For f in H^∞, f(T) can be defined without reference to the unitary dilation.

In fact if

${\displaystyle {\displaystyle f(z)=\sum _{n\ge 0}{a}_n{z}^n}}$

for |z| < 1, then for r < 1

${\displaystyle {\displaystyle f_r(z))=\sum _{n\ge 0}{r}^n{a}_n{z}^n}}$

is holomorphic on |z| < 1/r.

In that case f_r(T) is defined by the holomorphic functional calculus and f (T ) can be defined by

${\displaystyle {\displaystyle f(T)\xi =\underset{r\to 1}{lim}{f}_r(T)\xi .}}$

The map sending f to f(T) defines an algebra homomorphism of H^∞ into bounded operators on H. Moreover, if

${\displaystyle {\displaystyle f^{\sim }(z)=\sum _{n\ge 0}{a}_n{\overline{z}}^n,}}$

then

${\displaystyle {\displaystyle f^{\sim }(T)=f(T^{\ast }{)}^{\ast }.}}$

This map has the following continuity property: if a uniformly bounded sequence f_n tends almost everywhere to f, then f_n(T) tends to f(T) in the strong operator topology.

For t ≥ 0, let e_t be the inner function

${\displaystyle {\displaystyle e_t(z)=\exp t\frac{z+1}{z-1}.}}$

If T is the cogenerator of a one-parameter semigroup of completely non-unitary contractions T(t), then

${\displaystyle {\displaystyle T(t)=e_t(T)}}$

and

${\displaystyle {\displaystyle T=\frac{1}{2}I-\frac{1}{2}{\int }_0^{\mathrm{\infty }}{e}^{-t}T(t)dt.}}$

C0 contractions

A completely non-unitary contraction T is said to belong to the class C₀ if and only if f(T) = 0 for some non-zero f in H^∞. In this case the set of such f forms an ideal in H^∞. 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

${\displaystyle {\displaystyle \phi (z)=cB(z)e^{-P(z)},}}$

where |c|=1, B(z) is a Blaschke product

${\displaystyle {\displaystyle B(z)=\prod {\left[\frac{|{\lambda }_i|}{{\lambda }_i}\frac{{\lambda }_i-z}{1-{\overline{\lambda }}_i}\right]}^{m_i},}}$

with

${\displaystyle {\displaystyle \sum m_i(1-|{\lambda }_i|)<\mathrm{\infty },}}$

and P(z) is holomorphic with non-negative real part in D. By the Herglotz representation theorem,

${\displaystyle {\displaystyle P(z)={\int }_0^{2\pi }\frac{1+e^{-i\theta }z}{1-e^{-i\theta }z}d\mu (\theta )}}$

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. In the above decomposition of φ, either of the two factors can be absent.

The minimal function φ determines the spectrum of T. Within the unit disk, the spectral values are the zeros of φ. There are at most countably many such λ_i, all eigenvalues of T, the zeros of B(z). A point of the unit circle does not lie in the spectrum of T if and only if φ has a holomorphic continuation to a neighborhood of that point.

φ reduces to a Blaschke product exactly when H equals the closure of the direct sum (not necessarily orthogonal) of the generalized eigenspaces

${\displaystyle {\displaystyle H_i=\{\xi :(T-{\lambda }_{i}I{)}^{m_i}\xi =0\}.}}$

Quasi-similarity

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

${\displaystyle {\displaystyle A{T}_1=T_{2}A,B{T}_2=T_{1}B.}}$

The following properties of a contraction T are preserved under quasi-similarity:

• being unitary
• being completely non-unitary
• being in the class C₀
• being multiplicity free, i.e. having a commutative commutant

Two quasi-similar C₀ contractions have the same minimal function and hence the same spectrum.

The classification theorem for C₀ contractions states that two multiplicity free C₀ contractions are quasi-similar if and only if they have the same minimal function (up to a scalar multiple).

A model for multiplicity free C₀ contractions with minimal function φ is given by taking

${\displaystyle {\displaystyle H=H^2\ominus \phi H^2,}}$

where H² is the Hardy space of the circle and letting T be multiplication by z.

Such operators are called Jordan blocks and denoted S(φ).

As a generalization of Beurling's theorem, the commutant of such an operator consists exactly of operators ψ(T) with ψ in H^≈, i.e. multiplication operators on H² corresponding to functions in H^≈.

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).

Examples.

• If a contraction T if quasi-similar to an operator S with

${\displaystyle {\displaystyle S{e}_i={\lambda }_i{e}_i}}$

with the λ_i's distinct, of modulus less than 1, such that

${\displaystyle {\displaystyle \sum (1-|{\lambda }_i|)<1}}$

and (e_i) is an orthonormal basis, then S, and hence T, is C₀ and multiplicity free. Hence H is the closure of direct sum of the λ_i-eigenspaces of T, each having multiplicity one. This can also be seen directly using the definition of quasi-similarity.

• The results above can be applied equally well to one-parameter semigroups, since, from the functional calculus, two semigroups are quasi-similar if and only if their cogenerators are quasi-similar.

Classification theorem for C₀ contractions: Every C₀ contraction is canonically quasi-similar to a direct sum of Jordan blocks.

In fact every C₀ contraction is quasi-similar to a unique operator of the form

${\displaystyle {\displaystyle S=S({\phi }_1)\oplus S({\phi }_1{\phi }_2)\oplus S({\phi }_1{\phi }_2{\phi }_3)\oplus \cdots }}$

where the φ_n are uniquely determined inner functions, with φ₁ the minimal function of S and hence T.