View source for Contraction (operator theory)

{{Short description|Bounded operators with sub-unit norm}}
In [[operator theory]], a [[bounded operator]] ''T'': ''X'' → ''Y'' between [[normed vector space]]s ''X'' and ''Y'' is said to be a '''contraction''' if its [[operator norm]] ||''T'' ||&nbsp;≤&nbsp;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 ==
{{redirect|Operator angle||Camera angle}}

If ''T'' is a contraction acting on a [[Hilbert space]] <math>\mathcal{H}</math>, the following basic objects associated with ''T'' can be defined.

The '''defect operators''' of ''T'' are the operators ''D<sub>T</sub>''&nbsp;=&nbsp;(1&nbsp;&minus;&nbsp;''T*T'')<sup>½</sup> and ''D<sub>T*</sub>''&nbsp;=&nbsp;(1&nbsp;&minus;&nbsp;''TT*'')<sup>½</sup>. The square root is the [[square root of a matrix|positive semidefinite one]] given by the [[spectral theorem]]. The '''defect spaces''' <math>\mathcal{D}_T</math> and <math>\mathcal{D}_{T*}</math> are the closure of the ranges Ran(''D<sub>T</sub>'') and Ran(''D<sub>T*</sub>'') respectively. The positive operator ''D<sub>T</sub>'' induces an inner product on <math>\mathcal{H}</math>. The inner product space  can be identified naturally with Ran(''D''<sub>''T''</sub>). A similar statement holds for <math>\mathcal{D}_{T*}</math>.

The '''defect indices''' of ''T'' are the pair

:<math>(\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>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 subspace]]s on which its restriction is unitary. If ''U''&nbsp;=&nbsp;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&thinsp;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''<sup>''n''</sup> = ''P'' ''U''<sup>''n''</sup> ''P'' for all ''n'' > 0. The operator ''U'' is called a [[dilation (operator theory)|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<ref>{{harvnb|Sz.-Nagy|Foias|Bercovici|Kérchy|2010|pp=10–14}}</ref>

:<math>\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>\mathcal H</math> defined by

:<math>\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{\mathcal{K}=\mathcal{H} \oplus \mathcal{H}.}</math>

Define a unitary ''W'' on <math>\mathcal K</math> by

:<math>\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>\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.<ref>{{harvnb|Sz.-Nagy|Foias|Bercovici|Kérchy|2010|pp=24–28}}</ref>

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{\Phi(g)=PU(g)P,}</math>

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

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

where

:<math>\displaystyle{T=\sum \lambda_i U(g_i).}</math>

Moreover,

:<math>\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<sub>g</sub> v'', ''v''〉 where ''U<sub>g</sub>'' 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 <math>\mathcal H</math> be the space of functions on ''G'' of finite support with values in ''H'' with inner product

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

''G'' acts unitarily on <math>\mathcal H</math> by

:<math>\displaystyle{U(g)f(x)=f(g^{-1}x).}</math>

Moreover, ''H'' can be identified with a closed subspace of <math>\mathcal H</math> using the isometric embedding
sending ''v'' in ''H'' to ''f''<sub>''v''</sub> with

:<math>f_v(g)=\delta_{g,1} v. \, </math>

If ''P'' is the projection of <math>\mathcal H</math> onto ''H'', then

:<math>\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>\mathcal H</math>, so that <math>\mathcal H</math> is separable.

When ''G'' = '''Z''' any contraction operator ''T'' defines such a function Φ through

:<math>\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''.  {{harvtxt|Cooper|1947}} had previously proved the result for one-parameter semigroups of isometries,<ref>{{harvnb|Sz.-Nagy|Foias|Bercovici|Kérchy|2010|pp=28–30}}</ref>

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{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{\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{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{-\Re (A\xi,\xi)\ge 0}</math>

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

:<math>\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{T=(A+I)(A-I)^{-1}.}</math>

''A'' can be recovered from ''T'' using the formula

:<math>\displaystyle{A=(T+I)(T-I)^{-1}.}</math>

In particular a dilation of ''T'' on ''K'' ⊃ ''H'' immediately gives a dilation of the semigroup.<ref>{{harvnb|Sz.-Nagy|Foias|Bercovici|Kérchy|2010|pp=143, 147}}</ref>

==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<sup>2</sup>(''S''<sup>1</sup>).<ref>{{harvnb|Sz.-Nagy|Foias|Bercovici|Kérchy|2010|pp=87–88}}</ref>

If ''P'' is the orthogonal projection onto ''H'' then for ''f'' in L<sup>∞</sup> = L<sup>∞</sup>(''S''<sup>1</sup>) it follows that the operator ''f''(''T'') can be defined
by

:<math>\displaystyle{f(T)\xi=Pf(U)\xi.}</math>

Let H<sup>∞</sup> be the space of bounded holomorphic functions on the unit disk ''D''. Any such function has boundary values in L<sup>∞</sup> and is uniquely determined by these, so that there is an embedding  H<sup>∞</sup> ⊂  L<sup>∞</sup>.

For ''f'' in H<sup>∞</sup>, ''f''(''T'') can be defined
without reference to the unitary dilation.

In fact if

:<math>\displaystyle{f(z)=\sum_{n\ge 0} a_n z^n}</math>

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

:<math>\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''<sub>''r''</sub>(''T'') is defined by the holomorphic functional calculus and ''f'' (''T'' ) can be defined by

:<math>\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<sup>∞</sup> into bounded operators on ''H''. Moreover, if

:<math>\displaystyle{f^\sim(z)=\sum_{n\ge 0} a_n \overline{z}^n,}</math>

then

:<math>\displaystyle{f^\sim(T)=f(T^*)^*.}</math>

This map has the following continuity property: if a uniformly bounded sequence ''f''<sub>''n''</sub> tends almost everywhere to ''f'', then ''f''<sub>''n''</sub>(''T'') tends to ''f''(''T'') in the strong operator topology.

For ''t'' ≥ 0, let ''e''<sub>''t''</sub> be the inner function

:<math>\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{T(t)=e_t(T)}</math>

and

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

==C<sub>0</sub> contractions==
A completely non-unitary contraction ''T'' is said to belong to the class C<sub>0</sub> if and only if ''f''(''T'') = 0 for some non-zero
''f'' in H<sup>∞</sup>. In this case the set of such ''f'' forms an ideal in H<sup>∞</sup>. It has the form  φ ⋅ H<sup>∞</sup> where ''g'' 
is an [[inner function]], i.e. such that |φ| = 1 on ''S''<sup>1</sup>: 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 (linear algebra)|minimal polynomial]] of a matrix.

The minimal function φ admits a canonical factorization

:<math>\displaystyle{\varphi(z) = c B(z) e^{-P(z)},}</math>

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

:<math>\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{\sum m_i(1-|\lambda_i|) <\infty,}</math>
 
and ''P''(''z'') is holomorphic with non-negative real part in ''D''. By the [[positive harmonic function#Herglotz representation theorem for holomorphic functions|Herglotz representation theorem]],

:<math>\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 measure|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<sub>i</sub>, 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<ref>{{harvnb|Sz.-Nagy|Foias|Bercovici|Kérchy|2010|p=138}}</ref>

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

==Quasi-similarity==
Two contractions ''T''<sub>1</sub> and ''T''<sub>2</sub> are said to be '''quasi-similar''' when there are bounded operators ''A'', ''B'' with trivial kernel and dense range such that

:<math>\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<sub>0</sub>
*being '''multiplicity free''', i.e. having a commutative [[commutant]]

Two quasi-similar C<sub>0</sub> contractions have the same minimal function and hence the same spectrum.

The '''classification theorem''' for C<sub>0</sub> contractions states that two multiplicity free C<sub>0</sub> contractions are quasi-similar if and only if they have the same minimal function (up to a scalar multiple).<ref>{{harvnb|Sz.-Nagy|Foias|Bercovici|Kérchy|2010|pp=395–440}}</ref>

A model for multiplicity free C<sub>0</sub> contractions with minimal function φ is given by taking

:<math> \displaystyle{H=H^2\ominus \varphi H^2,}</math>

where H<sup>2</sup> is the [[Hardy space]] of the circle and letting ''T'' be multiplication by ''z''.<ref>{{harvnb|Sz.-Nagy|Foias|Bercovici|Kérchy|2010|p=126}}</ref>

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''<sup>≈</sup>, i.e. multiplication operators on ''H''<sup>2</sup> corresponding to functions in ''H''<sup>≈</sup>.
 
A C<sub>0</sub> 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{Se_i=\lambda_i e_i}</math>

with the λ<sub>i</sub>'s distinct, of modulus less than 1, such that

:<math>\displaystyle{\sum (1-|\lambda_i|) < 1}</math>

and (''e''<sub>''i''</sub>) is an orthonormal basis, then ''S'', and hence ''T'', is C<sub>0</sub> and multiplicity free.  Hence ''H'' is the closure of direct sum of the  λ<sub>i</sub>-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.<ref>{{harvnb|Bercovici|1988|p=95}}</ref>

'''Classification theorem for C<sub>0</sub> contractions:'''  ''Every C<sub>0</sub> contraction is canonically quasi-similar to a direct sum of Jordan blocks.''

In fact every C<sub>0</sub> contraction is quasi-similar to a unique operator of the form

:<math>\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<sub>''n''</sub> are uniquely determined inner functions, with φ<sub>''1''</sub> the minimal function of ''S'' and hence ''T''.<ref>{{harvnb|Bercovici|1988|pp=35–66}}</ref>

==See also==

* {{annotated link|Contraction mapping}}
* {{annotated link|Kallman–Rota inequality}}
* {{annotated link|Stinespring dilation theorem}}
* [[Hille-Yosida theorem#Hille-Yosida theorem for contraction semigroups|Hille-Yosida theorem for contraction semigroups]]

==Notes==
{{reflist|2}}

==References==

*{{citation|last=Bercovici|first= H.|title=Operator theory and arithmetic in H<sup>∞</sup>|series=Mathematical Surveys and Monographs|volume= 26|publisher= American Mathematical Society|year= 1988|isbn= 0-8218-1528-8}} 
*{{citation|last=Cooper|first=J. L. B.|title= One-parameter semigroups of isometric operators in Hilbert space|journal= Ann. of Math. |volume=48|year=1947|issue=4| pages=827–842|authorlink=Lionel Cooper (mathematician)|doi=10.2307/1969382|jstor=1969382}}
*{{citation|last=Gamelin|first= T. W.|title=Uniform algebras|publisher=Prentice-Hall|year= 1969}}
*{{citation|last=Hoffman|first= K.|title=Banach spaces of analytic functions|publisher=Prentice-Hall|year= 1962}}
*{{citation|last1=Sz.-Nagy|first1= B.|last2= Foias|first2= C.|last3= Bercovici|first3= H.|last4= Kérchy|first4= L.|
title=Harmonic analysis of operators on Hilbert space|edition=Second|series= Universitext|publisher= Springer|year= 2010|isbn= 978-1-4419-6093-1}} 
*{{citation|last1=Riesz|first1= F.|last2=Sz.-Nagy|first2= B.|title=Functional analysis. Reprint of the 1955 original|series= Dover Books on Advanced Mathematics|publisher=Dover|year= 1995|pages=466–472|isbn= 0-486-66289-6}}

{{Functional analysis}}
{{Banach spaces}}
{{Hilbert space}}

[[Category:Operator theory]]