A model for commuting pairs of contractions on Hilbert space - PowerPoint PPT Presentation

A model for commuting pairs of contractions on Hilbert space Nicholas Young Leeds and Newcastle Universities Newcastle, March 2017

Synopsis • Contractions and shifts • The Nagy-Foias functional model of a contraction • The symmetrized bidisc Γ – an interesting set in C 2 • Γ-analogues of contractions, unitaries and isometries • A Nagy-Foias functional model for Γ-contractions

The idea of a model of an operator There is no ‘canonical form’ of bounded linear operators on Hilbert space under unitary equivalence. The next best thing would be to express the general oper- ator as a part of a well-understood operator having a rich structure. This is the purpose of the book Harmonic analysis of operators on Hilbert space , by C. Foias and B. Sz.-Nagy, 1966. It expresses a general operator, up to unitary equivalence, in terms of ‘shift operators’ on Hilbert spaces of functions having analytic structure.

Contractions A contraction is a linear operator P on a Hilbert space H such that � P � ≤ 1. If P is a bounded linear operator on H then P is a contraction if id H − P ∗ P ≥ 0. The defect operator D P of a contraction P on H is the 1 positive operator (1 − P ∗ P ) 2 , acting on H . 1 The defect operator of P ∗ is thus D P ∗ = (1 − PP ∗ ) 2 . The defect space D P of P is ran D P , a subspace of H . We have P D P ⊂ D P ∗ .

Some basic objects Let E be a separable Hilbert space. L 2 ( E ) is the Lebesgue space of square-integrable E -valued functions f on T with norm � � f � 2 = T | f ( z ) | 2 m ( dz ) where m is normalized Lebesgue measure on T . H 2 ( E ) is the Hardy space of analytic E -valued functions on D . H 2 ( E ) can be thought of as the closed subspace of L 2 ( E ) comprising the functions f whose negative Fourier coeffi- cients ˆ f ( n ) , n < 0, vanish.

The shift operator The shift operator on H 2 ( E ) is the operator T z defined for f ∈ H 2 ( E ) by ( T z f )( z ) = zf ( z ) for z ∈ T . Note that T z shifts the Taylor coefficients of a function in H 2 ( E ): T z ( a 0 + a 1 z + a 2 z 2 + . . . ) = 0 + a 0 z + a 1 z 2 + . . . . The adjoint operator T ∗ z on H 2 ( E ) shifts coefficients back- wards : z ( a 0 + a 1 z + a 2 z 2 + . . . ) = a 1 + a 2 z + a 3 z 2 + . . . . T ∗ T z and T ∗ z are well understood operators.

The Nagy-Foias functional model of a contraction

An embedding of H in H 2 ( D P ∗ ) For a contraction P on H , define W : H → H 2 ( D P ∗ ) by Wx ∼ ( D P ∗ x, D P ∗ P ∗ x, D P ∗ ( P ∗ ) 2 x, . . . ) , Wx ( z ) = D P ∗ (1 − zP ∗ ) − 1 x. For any x ∈ H , � Wx � 2 = � x � 2 − lim N →∞ � ( P ∗ ) N x � 2 . Hence, if P is a ‘pure contraction’, meaning that ( P ∗ ) N tends strongly to zero on H as N → ∞ , then W is an iso- metric embedding of H in H 2 ( D P ∗ ). Moreover WP ∗ x = ( D P ∗ P ∗ x, D P ∗ ( P ∗ ) 2 x, . . . ) = T ∗ z ( D P ∗ x, D P ∗ P ∗ x, . . . ) = T ∗ z Wx where T z denotes the shift operator on H 2 ( D P ∗ ).

A functional model - part 1 Suppose P is a pure contraction. Let E denote the range of W . E is an invariant subspace of H 2 ( D P ∗ ) with respect to T ∗ z . Let U : H → E be the range restriction of W . Thus U is a unitary operator, and P = U ∗ ( T ∗ z |E ) ∗ U = U ∗ ( the compression of T z to E ) U. We need an effective description of the space E = ran W in terms of the operator P .

The characteristic operator function Θ P This is the analytic operator-valued function on D given by Θ P ( λ ) = [ − P + λD P ∗ (1 − λP ∗ ) − 1 D P ] |D P for λ ∈ D . Its values are contractive operators from D P to D P ∗ , and ran W = H 2 ( D P ∗ ) ⊖ Θ P H 2 ( D P ) . The functional model for pure contractions If P is a pure contraction then P is unitarily equivalent to the compression of the shift operator T z on H 2 ( D P ∗ ) to its co-invariant subspace H 2 ( D P ∗ ) ⊖ Θ P H 2 ( D P ) .

Completely non-unitary contractions Consider a contraction P on a Hilbert space H . If P has a nonzero invariant subspace H 1 on which it is isometric then P splits up into a block operator � � P 1 0 P = 0 P 2 with repect to the orthogonal decomposition H 1 ⊕ H ⊥ 1 of H . Here P 1 is a unitary operator, and so, by the Spectral Theorem, � P 1 = T λE (d λ ) for some spectral measure E . P 2 is a c.n.u. contraction : it has no nonzero unitary restric- tion.

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

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

Commuting pairs of contractions Let A and B be c.n.u contractions on a Hilbert space H such that AB = BA . A and B have Nagy-Foias models, but they typically act on different Hilbert spaces. No canonical model is known for the pair ( A, B ). Instead, we exhibit a canonical model for the symmetrization of ( A, B ), meaning the pair of operators ( A + B, AB ). ( A + B, AB ) is called a Γ -contraction .

The symmetrized bidisc: an interesting set in C 2

The symmetrized bidisc The closed symmetrized bidisc is the set Γ def = { ( z + w, zw ) : | z | ≤ 1 , | w | ≤ 1 } . Γ is a non-convex, polynomially convex set in C 2 . Γ is starlike about 0 but not circled. Γ ∩ R 2 is an isosceles triangle together with its interior. The distinguished boundary of Γ is the set b Γ def = { ( z + w, zw ) : | z | = | w | = 1 } , which is homeomorphic to the M¨ obius band.

Magic functions Define a rational function Φ z ( s, p ) of complex numbers z, s, p by Φ z ( s, p ) = 2 zp − s 2 − zs . For any z ∈ D , Φ z maps Γ into D − . Conversely, if ( s, p ) ∈ C 2 is such that | Φ z ( s, p ) | ≤ 1 for all z ∈ D then ( s, p ) ∈ Γ. This observation gives an analytic criterion for membership of Γ.

Γ -analogues of contractions, unitaries and isometries

Γ -contractions A Γ -contraction is a commuting pair ( S, P ) of bounded linear operators (on a Hilbert space H ) for which the symmetrized bidisc Γ def = { ( z + w, zw ) : | z | ≤ 1 , | w | ≤ 1 } is a spectral set. This means that, for all scalar polynomials g in two variables, � g ( S, P ) � ≤ sup | g | . Γ If ( S, P ) is a Γ-contraction then � S � ≤ 2 and � P � ≤ 1 (take g to be a co-ordinate functional). If A, B are commuting contractions then ( A + B, AB ) is a Γ-contraction, by Ando’s inequality.

Examples of Γ -contractions If ( S, P ) is a commuting pair of operators, then ( S, P ) has the form ( A + B, AB ) if and only if S 2 − 4 P is the square of an operator which commutes with S and P . If P is a contraction which has no square root then (0 , P ) is a Γ-contraction that is not of the form ( A + B, AB ) ( S, 0) is a Γ-contraction if and only if w ( S ) ≤ 1, where w is the numerical radius. The pair ( T z 1 + z 2 , T z 1 z 2 ) of analytic Toeplitz operators on H 2 ( D 2 ), restricted to the subspace H 2 sym of symmetric func- tions, is a Γ-contraction that is not of the form ( A + B, AB ).

A characterization of Γ -contractions For operators S, P let ρ ( S, P ) = 1 2 [(2 − S ) ∗ (2 − S ) − (2 P − S ) ∗ (2 P − S )] = 2(1 − P ∗ P ) − S + S ∗ P − S ∗ + P ∗ S. Theorem A commuting pair of operators ( S, P ) is a Γ-contraction if and only if ρ ( αS, α 2 P ) ≥ 0 for all α ∈ D . Necessity: for α ∈ D , Φ α is analytic on a neighbourhood of Γ and | Φ α | ≤ 1 on Γ. Hence, if ( S, P ) is a Γ-contraction, � ∗ � 2 αP − S � 2 αP − S � = 1 − Φ α ( S, P ) ∗ Φ α ( S, P ) ≥ 0 . 1 − 2 − αS 2 − αS

Γ -unitaries For a commuting pair ( S, P ) of operators on H the following statements are equivalent: (1) S and P are normal operators and the joint spectrum σ ( S, P ) lies in the distinguished boundary of Γ; (2) P ∗ P = 1 = PP ∗ and P ∗ S = S ∗ and � S � ≤ 2; (3) S = U 1 + U 2 and P = U 1 U 2 for some commuting pair of unitaries U 1 , U 2 on H . Define a Γ-unitary to be a commuting pair ( S, P ) for which (1)-(3) hold.

Γ -unitary dilations (Agler-Y, 1999, 2000) Theorem Every Γ -contraction has a Γ -unitary dilation. That is, if ( S, P ) is a Γ-contraction on H then there exist Hilbert spaces G ∗ , G and a Γ-unitary (˜ S, ˜ P ) on G ∗ ⊕ H ⊕ G having block operator matrices of the forms     ∗ 0 0 ∗ 0 0 ˜ ˜ S ∼ ∗ S 0  , P ∼ ∗ P 0  .       ∗ ∗ ∗ ∗ ∗ ∗ For any polynomial f in two variables, f ( S, P ) is the com- pression to H of f (˜ S, ˜ P ). Thus (˜ S, ˜ P ) is a dilation of ( S, P ).