A functional model for commuting pairs of contractions and the symmetrized bidisc

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1 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

2 Synopsis The Nagy-Foias functional model expresses the general completely 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 function of the operator. 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}.

3 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 f(z) = n=0 with a n E and f <, where f 2 = a n z n, z D, n=0 a n 2 E. L 2 (E) denotes the Lebesgue space of square-integrable E- valued functions on the circle.

4 The shift operator The shift operator T z on H 2 (E) is defined by 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 or equivalently T z f(z) = f(z) f(0) z 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?

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

6 An embedding of H in H 2 (D T ) Define a map W : H H 2 (D T ) by W x (D T x, D T T x, D T (T ) 2 x,... ), W x(z) = D T (1 zt ) 1 x. Then, for any x H, W x 2 = D T x 2 + D T T x = D T x, D T x + D T T x, D T T x +... = (1 T T )x, x + (1 T T )T x, T x +... = x 2 T x 2 + T x 2 (T ) 2 x 2 + (T ) 2 x 2... = lim N x 2 (T ) N x 2. 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 ).

7 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, W T x = (D T T x, D T (T ) 2 x,... ) = T z (D T x, D T T x,... ) = T z W x 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.

8 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 = W x. 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.

9 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.

10 An identity If M ΘT : H 2 (D T ) H 2 (D T ) is the operation of pointwise multiplication by Θ T then W W + M ΘT M Θ T = 1 H 2 (D T ). Proof. Let k be the Szegő kernel: k w (z) = (1 wz) 1 for z, w D. For any w D, x H and y D T, W (k w y), x H = k w y, W x 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 = (1 wt ) 1 D T y, x D T. Hence W (k w y) = (1 wt ) 1 D T y.

11 Proof of the identity continued Hence [ W W (k w y) ] (z) = [ W (1 wt ) 1 D T y ] (z) = D T (1 zt ) 1 (1 wt ) 1 D T y = (1 wz) 1 [ 1 Θ T (z)θ T (w) ]. Now [M ΘT MΘ 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 [W W + M ΘT MΘ T ](k w y) = k w y, and so W W + M ΘT MΘ = 1 T H 2 (D T ).

12 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 ). Proof. Since W is an isometry, W W is the orthogonal projection onto ran W. Hence M ΘT M Θ T = 1 H 2 (D T ) W W is the projection on (ran W ). Therefore ran W = (Θ T H 2 ).

13 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 ).

14 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.

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

16 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 T = [ H 2 (D T ) T L 2 (D T ) ] { Θ T u T u : u H 2 (D 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 Θ T (λ) = [ T + λd T (1 T λ) 1 ] D T DT. T is the Nagy-Foias model of T. It is canonical.

17 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.

18 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 V 0 0 T 0 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. Theorem (B. Sz.-Nagy, 1953) Every contraction on a Hilbert space has a unitary dilation. The minimal unitary dilation of a contraction is unique up to unitary equivalence..

19 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, U j 0 0 T j 0.

20 A drawback to Ando s theorem The simultaneous unitary dilation provided by Ando s theorem is not canonical. That is, not all such dilations (U 1, U 2 ) of a given pair (T 1, T 2 ) (even minimal) are unitarily equivalent. This fact suggests that there might not be a canonical functional model for commuting pairs of contractions. In the next lecture we shall see a slight modification of viewpoint which circumvents this obstacle. We shall introduce the notion of a Γ-contraction, where Γ is the set Γ = {(z + w, zw) : z 1, w 1}.

21 References [1] B. Sz.-Nagy and C. Foias, Harmonic Analysis of operators on Hilbert space, Akadémiai Kiadő, Budapest [2] N. K. Nikolskii and V. I. Vasyunin, Elements of spectral theory in terms of the free function model, Part I: Basic constructions, in Holomorphic Spaces, ed. S. Axler, J. E. McCarthy and D. Sarason, MSRI Publications, Berkeley, 1998, pages