# Contraction (operator theory)

__: Bounded operators with sub-unit norm__

**Short description**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. 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 [math]\displaystyle{ \mathcal{H} }[/math], 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*= (1 −

_{T*}*TT**)

^{½}. The square root is the positive semidefinite one given by the spectral theorem. The

**defect spaces**[math]\displaystyle{ \mathcal{D}_T }[/math] and [math]\displaystyle{ \mathcal{D}_{T*} }[/math] are the closure of the ranges Ran(

*D*) and Ran(

_{T}*D*) respectively. The positive operator

_{T*}*D*induces an inner product on [math]\displaystyle{ \mathcal{H} }[/math]. The inner product space can be identified naturally with Ran(

_{T}*D*

_{T}). A similar statement holds for [math]\displaystyle{ \mathcal{D}_{T*} }[/math].

The **defect indices** of *T* are the pair

- [math]\displaystyle{ (\dim\mathcal{D}_T, \dim\mathcal{D}_{T^*}). }[/math]

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

- [math]\displaystyle{ T = \Gamma \oplus U }[/math]

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^{[1]}

- [math]\displaystyle{ \displaystyle{\mathcal{H}=H\oplus H\oplus H \oplus \cdots ,} }[/math]

the orthogonal direct sum of countably many copies of *H*.

Let *V* be the isometry on [math]\displaystyle{ \mathcal H }[/math] defined by

- [math]\displaystyle{ \displaystyle{V(\xi_1,\xi_2,\xi_3,\dots)=(T\xi_1, \sqrt{I-T^*T}\xi_1,\xi_2,\xi_3,\dots).} }[/math]

Let

- [math]\displaystyle{ \displaystyle{\mathcal{K}=\mathcal{H} \oplus \mathcal{H}.} }[/math]

Define a unitary *W* on [math]\displaystyle{ \mathcal K }[/math] by

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

*W* is then a unitary dilation of *T* with *H* considered as the first component of [math]\displaystyle{ \mathcal{H}\subset \mathcal{K} }[/math].

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.^{[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

- [math]\displaystyle{ \displaystyle{\Phi(g)=PU(g)P,} }[/math]

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

- [math]\displaystyle{ \sum \lambda_i\overline{\lambda_j} \Phi(g_j^{-1}g_i) = PT^*TP\ge 0, }[/math]

where

- [math]\displaystyle{ \displaystyle{T=\sum \lambda_i U(g_i).} }[/math]

Moreover,

- [math]\displaystyle{ \displaystyle{\Phi(1)=P.} }[/math]

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*is a (strongly continuous) unitary representation (see Bochner's theorem). Replacing

_{g}*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 [math]\displaystyle{ \mathcal H }[/math] be the space of functions on *G* of finite support with values in *H* with inner product

- [math]\displaystyle{ \displaystyle{(f_1,f_2)=\sum_{g,h} (\Phi(h^{-1}g)f_1(g),f_2(h)).} }[/math]

*G* acts unitarily on [math]\displaystyle{ \mathcal H }[/math] by

- [math]\displaystyle{ \displaystyle{U(g)f(x)=f(g^{-1}x).} }[/math]

Moreover, *H* can be identified with a closed subspace of [math]\displaystyle{ \mathcal H }[/math] using the isometric embedding
sending *v* in *H* to *f*_{v} with

- [math]\displaystyle{ f_v(g)=\delta_{g,1} v. \, }[/math]

If *P* is the projection of [math]\displaystyle{ \mathcal H }[/math] onto *H*, then

- [math]\displaystyle{ \displaystyle{PU(g)P=\Phi(g),} }[/math]

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 [math]\displaystyle{ \mathcal H }[/math], so that [math]\displaystyle{ \mathcal H }[/math] is separable.

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

- [math]\displaystyle{ \displaystyle \Phi(0)=I, \,\,\, \Phi(n)=T^n,\,\,\, \Phi(-n)=(T^*)^n, }[/math]

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,^{[3]}

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

- [math]\displaystyle{ \displaystyle{T(t)=PU(t)P} }[/math]

and the translates *U*(*t*)*H* generate *K*.

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

- [math]\displaystyle{ \displaystyle{\Phi(0)=I, \,\,\, \Phi(t)=T(t),\,\,\, \Phi(-t)= T(t)^*,} }[/math]

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

- [math]\displaystyle{ \displaystyle{A\xi=\lim_{t\downarrow 0} {1\over t}(T(t)-I)\xi,} }[/math]

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

*A* is called the **generator** of the semigroup and satisfies

- [math]\displaystyle{ \displaystyle{-\Re (A\xi,\xi)\ge 0} }[/math]

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

- [math]\displaystyle{ \displaystyle{T(t)=e^{At},} }[/math]

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

- [math]\displaystyle{ \displaystyle{T=(A+I)(A-I)^{-1}.} }[/math]

*A* can be recovered from *T* using the formula

- [math]\displaystyle{ \displaystyle{A=(T+I)(T-I)^{-1}.} }[/math]

In particular a dilation of *T* on *K* ⊃ *H* immediately gives a dilation of the semigroup.^{[4]}

## 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}).^{[5]}

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

- [math]\displaystyle{ \displaystyle{f(T)\xi=Pf(U)\xi.} }[/math]

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

- [math]\displaystyle{ \displaystyle{f(z)=\sum_{n\ge 0} a_n z^n} }[/math]

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

- [math]\displaystyle{ \displaystyle{f_r(z))=\sum_{n\ge 0} r^n a_n z^n} }[/math]

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

- [math]\displaystyle{ \displaystyle{f(T)\xi=\lim_{r\rightarrow 1} f_r(T)\xi.} }[/math]

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

- [math]\displaystyle{ \displaystyle{f^\sim(z)=\sum_{n\ge 0} a_n \overline{z}^n,} }[/math]

then

- [math]\displaystyle{ \displaystyle{f^\sim(T)=f(T^*)^*.} }[/math]

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

- [math]\displaystyle{ \displaystyle{e_t(z)=\exp t{z+1\over z-1}.} }[/math]

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

- [math]\displaystyle{ \displaystyle{T(t)=e_t(T)} }[/math]

and

- [math]\displaystyle{ \displaystyle{T={1\over 2}I -{1\over 2}\int_0^\infty e^{-t}T(t)\, dt.} }[/math]

## 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

- [math]\displaystyle{ \displaystyle{\varphi(z) = c B(z) e^{-P(z)},} }[/math]

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

- [math]\displaystyle{ \displaystyle{B(z)=\prod \left[{|\lambda_i|\over \lambda_i} {\lambda_i -z \over 1-\overline{\lambda}_i }\right]^{m_i},} }[/math]

with

- [math]\displaystyle{ \displaystyle{\sum m_i(1-|\lambda_i|) \lt \infty,} }[/math]

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

- [math]\displaystyle{ \displaystyle{P(z) =\int_0^{2\pi} {1 + e^{-i\theta}z\over 1 -e^{-i\theta}z} \, d\mu(\theta)} }[/math]

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^{[6]}

- [math]\displaystyle{ \displaystyle{H_i=\{\xi:(T-\lambda_i I)^{m_i} \xi=0\}.} }[/math]

## 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

- [math]\displaystyle{ \displaystyle{AT_1=T_2A,\,\,\, BT_2=T_1B.} }[/math]

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).^{[7]}

A model for multiplicity free C_{0} contractions with minimal function φ is given by taking

- [math]\displaystyle{ \displaystyle{H=H^2\ominus \varphi H^2,} }[/math]

where H^{2} is the Hardy space of the circle and letting *T* be multiplication by *z*.^{[8]}

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

- [math]\displaystyle{ \displaystyle{Se_i=\lambda_i e_i} }[/math]

with the λ_{i}'s distinct, of modulus less than 1, such that

- [math]\displaystyle{ \displaystyle{\sum (1-|\lambda_i|) \lt 1} }[/math]

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.
^{[9]}

**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

- [math]\displaystyle{ \displaystyle{S=S(\varphi_1)\oplus S(\varphi_1\varphi_2)\oplus S(\varphi_1\varphi_2\varphi_3) \oplus \cdots } }[/math]

where the φ_{n} are uniquely determined inner functions, with φ_{1} the minimal function of *S* and hence *T*.^{[10]}

## See also

- Kallman–Rota inequality
- Stinespring dilation theorem
- Hille-Yosida theorem for contraction semigroups

## Notes

- ↑ Sz.-Nagy et al. 2010, pp. 10–14
- ↑ Sz.-Nagy et al. 2010, pp. 24–28
- ↑ Sz.-Nagy et al. 2010, pp. 28–30
- ↑ Sz.-Nagy et al. 2010, pp. 143, 147
- ↑ Sz.-Nagy et al. 2010, pp. 87–88
- ↑ Sz.-Nagy et al. 2010, p. 138
- ↑ Sz.-Nagy et al. 2010, pp. 395–440
- ↑ Sz.-Nagy et al. 2010, p. 126
- ↑ Bercovici 1988, p. 95
- ↑ Bercovici 1988, pp. 35–66

## References

- Bercovici, H. (1988),
*Operator theory and arithmetic in H*, Mathematical Surveys and Monographs,^{∞}**26**, American Mathematical Society, ISBN 0-8218-1528-8 - Cooper, J. L. B. (1947), "One-parameter semigroups of isometric operators in Hilbert space",
*Ann. of Math.***48**(4): 827–842, doi:10.2307/1969382 - Gamelin, T. W. (1969),
*Uniform algebras*, Prentice-Hall - Hoffman, K. (1962),
*Banach spaces of analytic functions*, Prentice-Hall - Sz.-Nagy, B.; Foias, C.; Bercovici, H.; Kérchy, L. (2010),
*Harmonic analysis of operators on Hilbert space*, Universitext (Second ed.), Springer, ISBN 978-1-4419-6093-1 - Riesz, F.; Sz.-Nagy, B. (1995),
*Functional analysis. Reprint of the 1955 original*, Dover Books on Advanced Mathematics, Dover, pp. 466–472, ISBN 0-486-66289-6

Original source: https://en.wikipedia.org/wiki/Contraction (operator theory).
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