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 2 The symmetrized bidisc Γ and Γ -contractions St Petersburg, June 2016

Symmetrization The symmetrization map π is given by π ( z, w ) = ( z + w, zw ) . The closed symmetrized bidisc is the set Γ def = { ( z + w, zw ) : | z | ≤ 1 , | w | ≤ 1 } . For any commuting pair ( A, B ) of contractions on a Hilbert space H , we shall construct a canonical model of the sym- metrization of ( A, B ), that is, of π ( A, B ) = ( A + B, AB ). Let ( S, P ) = π ( A, B ). Then ( S, P ) is a commuting pair of operators on H with � S � ≤ 2 and � P � ≤ 1.

Ando’s inequality Let A, B be commuting contractions on H . The following is a consequence of (1) Ando’s theorem on the existence of a simultaneous unitary dilation of ( A, B ) and (2) the spectral theorem for commuting unitaries: For any polynomial f in two variables, � f ( A, B ) � ≤ sup D 2 | f | . If ( S, P ) = π ( A, B ), then for any polynomial g and f = g ◦ π , � g ( S, P ) � = � f ( A, B ) � ≤ sup D 2 | f | = sup D 2 | g ◦ π | = sup | g | . Γ That is, Γ is a spectral set for the pair ( S, P ).

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 the previous slide.

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 ).

Some properties of the symmetrized bidisc 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.

Characterizations of Γ The following statements are equivalent for ( s, p ) ∈ C 2 . (1) ( s, p ) ∈ Γ, that is, s = z + w and p = zw for some z, w ∈ D − ; sp | ≤ 1 − | p | 2 and | s | ≤ 2; (2) | s − ¯ sp | + | s 2 − 4 p | + | s | 2 ≤ 4; (3) 2 | s − ¯ (4) 2 zp − s � � � � � ≤ 1 for all z ∈ D . � � 2 − zs �

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 . By the last slide, 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.

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

A sketch of sufficiency Suppose that ρ ( αS, α 2 P ) ≥ 0 for all α ∈ D . Consider a polynomial g such that | g | ≤ 1 on G . By Ando’s Theorem, � g ( A + B, AB ) � ≤ 1 for all commuting pairs ( A, B ). Use this property to prove an integral representation formula for 1 − g ∗ g . There exist a Hilbert space E , a B ( E )-valued spectral measure E on T and a continuous function F : T × Γ → E (such that F ( ω, · ) is analytic on Γ for every ω ∈ T ) for which � ω 2 p ) � E ( dω ) F ( ω, s, p ) , F ( ω, s, p ) � 1 − g ( s, p ) g ( s, p ) = T ρ (¯ ωs, ¯ for all ( s, p ) ∈ Γ. Apply to the commuting pair ( S, P ); the right hand side is clearly positive. Thus � g ( S, P ) � ≤ 1.

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.

Do Γ -contractions have Γ -unitary dilations? Let ( S, P ) be a Γ-contraction on H . Then P is a contraction, and so P has a minimal unitary dilation ˜ P on a Hilbert space K ⊃ H . By the Commutant Lifting Theorem, there exists an oper- ator ˜ S on K which commutes with ˜ P , has norm � S � and is a dilation of S . It does not follow that (˜ S, ˜ P ) is a Γ-unitary, or even a Γ- contraction. Can we choose ˜ S so that (˜ S, ˜ P ) is a Γ-unitary?

Yes (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 ).

Outline of the proof 1 If Γ is a spectral set for a commuting The main Lemma pair ( S, P ) then Γ is a complete spectral set for ( S, P ). Let ( S, P ) be a Γ-contraction on H . Let P 2 be the algebra of polynomials in two variables, and for f ∈ P 2 let f ♯ ∈ C ( T 2 ) be defined by f ♯ ( z 1 , z 2 ) = f ( z 1 + z 2 , z 1 z 2 ) . The map f �→ f ♯ is an algebra-embedding of P 2 in C ( T 2 ) Let its range be P ♯ 2 . Define an algebra representation θ : P ♯ 2 → B ( H ) by θ ( f ♯ ) = f ( S, P ) .

Outline of the proof 2 The fact that Γ is a complete spectral set for ( S, P ) implies that θ is a completely contractive representation of the algebra P ♯ 2 ⊂ C ( T 2 ), on H . By Arveson’s Extension Theorem and Stinespring’s Theo- rem there is a Hilbert space K ⊃ H and a unital ∗ -representation Ψ : C ( T 2 ) → B ( K ) such that f ( S, P ) = θ ( f ♯ ) = P H Ψ( f ♯ ) |H for all polynomials f. The operators S def P def ˜ ˜ = Ψ( z 1 + z 2 ) , = Ψ( z 1 z 2 ) on K have the desired properties: (˜ S, ˜ P ) is a Γ-unitary dilation of ( S, P ).

Isometries For V ∈ B ( H ), the following statements are equivalent: (1) � V x � = � x � for all x ∈ H ; (2) V ∗ V = 1; (3) V = U |H for some unitary U on a superspace of H such that H is a U -invariant subspace. V is an isometry if (1)-(3) hold. V is a pure isometry if, in addition, there is no non-trivial reducing subspace of H on which V is unitary. A pure isometry V is unitarily equivalent to multiplication by z on H 2 ( E ), where E = ker V .

Γ -isometries Define a Γ-isometry to be the restriction of a Γ-unitary (˜ S, ˜ P ) to a joint invariant subspace of (˜ S, ˜ P ). For commuting operators S, P on a Hilbert space H the following statements are equivalent: (1) ( S, P ) is a Γ-isometry; (2) P ∗ P = 1 and P ∗ S = S ∗ and � S � ≤ 2; (3) � S � ≤ 2 and (2 − ωS ) ∗ (2 − ωS ) − (2 ωP − S ) ∗ (2 ωP − S ) ≥ 0 for all ω ∈ T . if ( S ∗ , P ∗ ) is a Γ-isometry. ( S, P ) is called a Γ -co-isometry

Pure Γ -isometries If ( S, P ) is a Γ-isometry and the isometry P is pure (i.e. has a trivial unitary part) then ( S, P ) is called a pure Γ-isometry. P , being a pure isometry, is unitarily equivalent to the for- ward shift operator (multiplication by z ) on the vectorial Hardy space H 2 ( E ), where E = ker P . Since S commutes with the shift, S is the operation of mul- tiplication by a bounded analytic B ( E )-valued function on H 2 ( E ).

A Wold decomposition for Γ -isometries Every isometry is the orthogonal direct sum of a unitary and a pure isometry (a forward shift operator) (Wold-Kolmogorov). Every Γ-isometry is the orthogonal direct sum of a Γ-unitary and a pure Γ-isometry. That is: Let ( S, P ) be a Γ-isometry on H . There exists an orthogonal decomposition H = H 1 ⊕ H 2 such that (1) H 1 , H 2 are reducing subspaces of both S and P , (2) ( S, P ) |H 1 is a Γ-unitary, (3) ( S, P ) |H 2 is a pure Γ-isometry.