A functional model for commuting pairs of contractions and the - PowerPoint PPT Presentation

A functional model for commuting pairs of contractions and the symmetrized bidisc Nicholas Young Leeds and Newcastle Universities Lecture 1 The Nagy-Foias model of a contractive operator St Petersburg, June 2016

Synopsis The Nagy-Foias functional model expresses the general com- pletely non-unitary contractive linear operator T on Hilbert space as a compression of the shift operator on an L 2 space of vector-valued analytic functions on the unit circle. The compression is to a semi-invariant subspace H T , which has a concrete description in terms of the characteristic operator of the operator. function The purpose of these lectures is to extend the Nagy-Foias functional model to pairs of commuting contractions, and indeed to a larger class, called Γ -contractions . To do this we need to study the symmetrized bidisc , which is the set Γ = { ( z + w, zw ) : | z | ≤ 1 , | w | ≤ 1 } .

Some basic objects Throughout this lecture T is a bounded linear operator of norm at most one on a separable Hilbert space H . Such an object is a contraction . For any separable Hilbert space E , we denote by H 2 ( E ) the space of analytic E -valued functions on D of the form ∞ a n z n , � f ( z ) = z ∈ D , n =0 with a n ∈ E and � f � < ∞ , where ∞ � f � 2 = � a n � 2 � E . n =0 L 2 ( E ) denotes the Lebesgue space of square-integrable E - valued functions on the circle.

The shift operator on H 2 ( E ) is defined by The shift operator T z T z f ( z ) = zf ( z ) , z ∈ D , or equivalently T z ( a 0 , a 1 , . . . ) = (0 , a 0 , a 1 , . . . ) . Its adjoint T ∗ z , the backward shift , is given by z f ( z ) = f ( z ) − f (0) T ∗ z or equivalently T ∗ z ( a 0 , a 1 , . . . ) = ( a 1 , a 2 , . . . ) . To what extent do T z and T ∗ z provide models for the general Hilbert space contraction?

Defect operators and spaces The defect operator D T of T is defined to be the positive 1 operator (1 − T ∗ T ) 2 , acting on H . The defect operator of T ∗ is thus 1 D T ∗ = (1 − TT ∗ ) 2 . Note that 1 1 TD T = T (1 − T ∗ T ) 2 = (1 − TT ∗ ) 2 T = D T ∗ T. The defect space D T of T is ran D T ⊂ H . We have T D T ⊂ D T ∗ .

An embedding of H in H 2 ( D T ∗ ) Define a map W : H → H 2 ( D T ∗ ) by Wx ∼ ( D T ∗ x, D T ∗ T ∗ x, D T ∗ ( T ∗ ) 2 x, . . . ) , Wx ( z ) = D T ∗ (1 − zT ∗ ) − 1 x. Then, for any x ∈ H , � Wx � 2 = � D T ∗ x � 2 + � D T ∗ T ∗ x � 2 + . . . = � D T ∗ x, D T ∗ x � + � D T ∗ T ∗ x, D T ∗ T ∗ x � + . . . = � (1 − TT ∗ ) x, x � + � (1 − TT ∗ ) T ∗ x, T ∗ x � + . . . = � x � 2 − � T ∗ x � 2 + � T ∗ x � 2 − � ( T ∗ ) 2 x � 2 + � ( T ∗ ) 2 x � 2 − . . . N →∞ � x � 2 − � ( T ∗ ) N x � 2 . = lim Hence, if ( T ∗ ) N tends strongly to zero on H as N → ∞ , then W is an isometric embedding of H in H 2 ( D T ∗ ).

An intertwining Say that T is a pure contraction if ( T ∗ ) N → 0 as N → ∞ . Let W : H → H 2 ( D T ∗ ) be the isometric embedding just constructed. Then, for x ∈ H , WT ∗ x = ( D T ∗ T ∗ x, D T ∗ ( T ∗ ) 2 x, . . . ) = T ∗ z ( D T ∗ x, D T ∗ T ∗ x, . . . ) = T ∗ z Wx where T z denotes the shift operator on H 2 ( D T ∗ ). Let E denote the range of W . Observe that E is an invariant subspace of H 2 ( D T ∗ ) with respect to T ∗ z .

Theorem If T is a pure contraction then there is a unitary map U from H to an T ∗ z -invariant subspace E of H 2 ( D T ∗ ) such that T ∗ = U ∗ ( T ∗ z |E ) U. Thus T = U ∗ ( T ∗ z |E ) ∗ U = U ∗ ( the compression of T z to E ) U. We have proved this theorem: just let U : H → E be given by Ux = Wx . This is already a ‘functional model’ of sorts, at least for pure contractions. The shift operator on H 2 ( D T ∗ ) can be analysed by means of classical function theory, whence the title of Nagy and Foias’ book, Harmonic analysis of operators on Hilbert space . To give the model power, we need an effective description of the space E = ran W in terms of the operator T .

The characteristic operator function Θ T This is the operator-valued function on D given by Θ T ( λ ) = [ − T + λD T ∗ (1 − λT ∗ ) − 1 D T ] |D T for λ ∈ D . Its values are contractive operators from D T to D T ∗ . Θ T is a purely contractive analytic function ,which means that: • Θ T is analytic on D , • its values are contractive operators, and • for every vector x ∈ D T , � Θ T (0) x � < � x � . Exercise: for z, w ∈ D , 1 − Θ T ( z )Θ T ( w ) ∗ = (1 − ¯ wz ) D T ∗ (1 − zT ∗ ) − 1 (1 − ¯ wT ) − 1 D T ∗ .

An identity If M Θ T : H 2 ( D T ) → H 2 ( D T ∗ ) is the operation of pointwise multiplication by Θ T then WW ∗ + M Θ T M ∗ Θ T = 1 H 2 ( D T ∗ ) . wz ) − 1 for Proof. Let k be the Szeg˝ o kernel: k w ( z ) = (1 − ¯ z, w ∈ D . For any w ∈ D , x ∈ H and y ∈ D T ∗ , � W ∗ ( k w ⊗ y ) , x � H = � k w ⊗ y, Wx � H 2 ( D T ∗ ) k w ⊗ y, D T ∗ (1 − zT ∗ ) − 1 x � � = H 2 ( D T ∗ ) � y, D T ∗ (1 − wT ∗ ) − 1 x � = D T ∗ � wT ) − 1 D T ∗ y, x � = (1 − ¯ D T ∗ . Hence W ∗ ( k w ⊗ y ) = (1 − ¯ wT ) − 1 D T ∗ y.

Proof of the identity continued Hence � ( z ) = � WW ∗ ( k w ⊗ y ) � wT ) − 1 D T ∗ y � W (1 − ¯ ( z ) = D T ∗ (1 − zT ∗ ) − 1 (1 − ¯ wT ) − 1 D T ∗ y wz ) − 1 � 1 − Θ T ( z )Θ T ( w ) ∗ � . = (1 − ¯ Now [ M Θ T M ∗ Θ T ( k w ⊗ y )]( z ) = Θ T ( z )[ k w ⊗ Θ T ( w ) ∗ y ]( z ) = k w ( z )Θ T ( z )Θ T ( w ) ∗ y. On adding these two equations we find [ WW ∗ + M Θ T M ∗ Θ T ]( k w ⊗ y ) = k w ⊗ y, and so WW ∗ + M Θ T M ∗ Θ T = 1 H 2 ( D T ∗ ) .

The range of W : H → H 2 ( D T ∗ ) Proposition. If T is a pure contraction then ran W = H 2 ( D T ∗ ) ⊖ Θ T H 2 ( D T ) where Θ T H 2 ( D T ) def = { Θ T f : f ∈ H 2 ( D T ) } ⊂ H 2 ( D T ∗ ) . Since W is an isometry, WW ∗ is the orthogonal Proof. projection onto ran W . Θ T = 1 H 2 ( D T ∗ ) − WW ∗ is the projection on Hence M Θ T M ∗ (ran W ) ⊥ . Therefore ran W = (Θ T H 2 ) ⊥ .

Theorem - the functional model for pure contractions If T is a pure contraction then T is unitarily equivalent to the compression of the shift operator T z on H 2 ( D T ∗ ) to its co-invariant subspace H 2 ( D T ∗ ) ⊖ Θ T H 2 ( D T ) .

Completely non-unitary contractions A c.n.u. contraction is a contraction which has no nontrivial unitary restriction. If T is a contraction on H then the set K = { x ∈ H : � T n x � = � x � = � ( T ∗ ) n � for n ≥ 1 } is a T -reducing subspace of H on which T acts as a unitary operator. It is clearly the largest such subspace. Thus T is the orthogonal direct sum of a unitary operator T |K and a c.n.u. contraction T |K ⊥ . Pure contractions are c.n.u. The functional model of the last slide can be extended to c.n.u. contractions.

The model space H T Let T be a c.n.u contraction on H . Define an operator- valued function on T by 1 ∆ T ( e it ) = [1 − Θ T ( e it ) ∗ Θ T ( e it )] 2 . For almost all t ∈ R , ∆ T ( e it ) is an operator on D T . The model space H T is defined by H 2 ( D T ∗ ) ⊕ ∆ T L 2 ( D T ) Θ T u ⊕ ∆ T u : u ∈ H 2 ( D T ) � � � � H T = ⊖ . H T is a space of functions on T with values in D T ∗ ⊕ D T .

Theorem: the Nagy-Foias functional model Let T be a c.n.u contraction on H . Then T is unitarily equivalent to the operator T on the model space H 2 ( D T ∗ ) ⊕ ∆ T L 2 ( D T ) Θ T u ⊕ ∆ T u : u ∈ H 2 ( D T ) � � � � H T = ⊖ given by T ∗ ( u ⊕ v ) = e − it [ u ( e it ) − u (0)] ⊕ e − it v ( e it ) for all u ⊕ v ∈ H T . Here � − T + λD T ∗ (1 − T ∗ λ ) − 1 D T � Θ T ( λ ) = |D T . T is the Nagy-Foias model of T . It is canonical.

Commuting pairs of contractions Let T 1 , T 2 be commuting contractions on a Hilbert space H Is there a canonical functional model of ( T 1 , T 2 ), of Nagy- Foias type? Claim: There is a canonical functional model of a Γ -contraction , (to be defined). This model can be interpreted as a model for a commuting pair of contractions. The first step is to construct a ‘canonical Γ-unitary dilation’.

Unitary dilations Let T be an operator on a Hilbert space H . Consider an operator V on the space G ∗ ⊕H⊕G (for some Hilbert spaces G ∗ , G ) of the form   ∗ 0 0 V ∼ ∗ 0 T  .    ∗ ∗ ∗ For any polynomial f , the compression of f ( V ) to H is f ( T ). An operator V (on a superspace of H ) with this property is called a dilation of T . (B. Sz.-Nagy, 1953) Every contraction on a Theorem Hilbert space has a unitary dilation. The minimal unitary dilation of a contraction is unique up to unitary equivalence.

Ando’s theorem A commuting pair T 1 , T 2 of commuting contractions on a Hilbert space H has a simultaneous unitary dilation. That is, there exists a superspace K of H and a commuting pair U 1 , U 2 of unitaries on K such that, for every polynomial f in two variables, f ( T 1 , T 2 ) is the compression to H of f ( U 1 , U 2 ). Equivalently, for some orthogonal decomposition G ∗ ⊕ H ⊕ G of K , and for j = 1 , 2,   ∗ 0 0 U j ∼ ∗ 0 T j  .    ∗ ∗ ∗