Contraction (operator theory)
# Contraction (operator theory)

In operator theory, a bounded operator T: X → Y between normed vector spaces X and Y is said to be a contraction if its operator norm ||T || ≤ 1. This notion is a special case of the concept of a contraction mapping, but every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into the structure of operators, or a family of operators. The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias.

## Contractions on a Hilbert space

If T is a contraction acting on a Hilbert space ${\mathcal {H}}$, the following basic objects associated with T can be defined.

The defect operators of T are the operators $D_T$ = (1 − $T*T)^½$ and $D_{T*}$ = (1 − $TT*)^½.$ The square root is the positive semidefinite one given by the spectral theorem. The defect spaces ${\mathcal {D}}_{T}$ and ${\mathcal {D}}_{T*}$ are the closure of the ranges $Ran(D_T)$ and $Ran(D_{T*})$ respectively. The positive operator $D_T$ induces an inner product on ${\mathcal {H}}$. The inner product space can be identified naturally with $Ran(D_T).$ A similar statement holds for ${\mathcal {D}}_{T*}$.

The defect indices of T are the pair

    $(\dim {\mathcal {D}}_{T},\dim {\mathcal {D}}_{T^{*}}).$

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

    $T=\Gamma \oplus U$

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. If U = 0, T is said to be a completely non-unitary contraction. A special case of this decomposition is the Wold decomposition for an isometry, where Γ is a proper isometry.

Contractions on Hilbert spaces can be viewed as the operator analogs of cos θ and are called operator angles in some contexts. The explicit description of contractions leads to (operator-)parametrizations of positive and unitary matrices.

## 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 $T^n$ = P $U^n$ P 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

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

the orthogonal direct sum of countably many copies of H.

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

    $\displaystyle {V(\xi _{1},\xi _{2},\xi _{3},\dots )=(T\xi _{1},{\sqrt {I-T^{*}T}}\xi _{1},\xi _{2},\xi _{3},\dots ).}$

Let

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

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

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

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

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

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

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

    $\sum \lambda _{i}{\overline {\lambda _{j}}}\Phi (g_{j}^{-1}g_{i})=PT^{*}TP\geq 0,$

where

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

Moreover,

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

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

    $\displaystyle {(f_{1},f_{2})=\sum _{g,h}(\Phi (h^{-1}g)f_{1}(g),f_{2}(h)).}$

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

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

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

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

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

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

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 ${\mathcal {H}}$, so that ${\mathcal {H}}$ is separable.

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

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

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

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

and the translates U(t)H generate K.

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

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

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

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

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

A is called the generator of the semigroup and satisfies

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

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

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

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

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

A can be recovered from T using the formula

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

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 $L^2(S^1).$

If P is the orthogonal projection onto H then for f in $L^∞$ = $L^∞(S^1)$ it follows that the operator f(T) can be defined by

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

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

    $\displaystyle {f(z)=\sum _{n\geq 0}a_{n}z^{n}}$

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

    $\displaystyle {f_{r}(z))=\sum _{n\geq 0}r^{n}a_{n}z^{n}}$

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

    $\displaystyle {f(T)\xi =\lim _{r\rightarrow 1}f_{r}(T)\xi .}$

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

    $\displaystyle {f^{\sim }(z)=\sum _{n\geq 0}a_{n}{\overline {z}}^{n},}$

then

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

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

    $\displaystyle {e_{t}(z)=\exp t{z+1 \over z-1}.}$

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

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

and

    $\displaystyle {T={1 \over 2}I-{1 \over 2}\int _{0}^{\infty }e^{-t}T(t)\,dt.}$

## $C_0$ contractions

A completely non-unitary contraction T is said to belong to the class $C_0$ 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 $S^1:$ 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

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

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

    $\displaystyle {B(z)=\prod \left[{|\lambda _{i}| \over \lambda _{i}}{\lambda _{i}-z \over 1-{\overline {\lambda }}_{i}}\right]^{m_{i}},}$

with

    $\displaystyle {\sum m_{i}(1-|\lambda _{i}|)<\infty ,}$

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

    $\displaystyle {P(z)=\int _{0}^{2\pi }{1+e^{-i\theta }z \over 1-e^{-i\theta }z}\,d\mu (\theta )}$

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

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

## Quasi-similarity

Two contractions $T_1$ and $T_2$ are said to be quasi-similar when there are bounded operators A, B with trivial kernel and dense range such that

    $\displaystyle {AT_{1}=T_{2}A,\,\,\,BT_{2}=T_{1}B.}$

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

* being unitary
* being completely non-unitary
* being in the class $C_0$
* being multiplicity free, i.e. having a commutative commutant

Two quasi-similar $C_0$ contractions have the same minimal function and hence the same spectrum.

The classification theorem for $C_0$ contractions states that two multiplicity free $C_0$ contractions are quasi-similar if and only if they have the same minimal function (up to a scalar multiple).

A model for multiplicity free $C_0$ contractions with minimal function φ is given by taking

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

where $H^2$ 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 $H^2$ corresponding to functions in $H^≈.$

A $C_0$ 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

    $\displaystyle {Se_{i}=\lambda _{i}e_{i}}$

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

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

and $(e_i)$ is an orthonormal basis, then S, and hence T, is $C_0$ 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 $C_0$ contractions: Every $C_0$ contraction is canonically quasi-similar to a direct sum of Jordan blocks.

In fact every $C_0$ contraction is quasi-similar to a unique operator of the form

    $\displaystyle {S=S(\varphi _{1})\oplus S(\varphi _{1}\varphi _{2})\oplus S(\varphi _{1}\varphi _{2}\varphi _{3})\oplus \cdots }$

where the $φ_n$ are uniquely determined inner functions, with $φ_1$ the minimal function of S and hence T.

