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

2 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 symmetrization 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.

3 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 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. f = sup D 2 That is, Γ is a spectral set for the pair (S, P ). g π = sup Γ g.

4 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 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. g.

5 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 4P 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 z1 +z 2, T z1 z 2 ) of analytic Toeplitz operators on H 2 (D 2 ), restricted to the subspace H 2 sym of symmetric functions, is a Γ-contraction that is not of the form (A+B, AB).

6 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 bγ def = {(z + w, zw) : z = w = 1}, which is homeomorphic to the Möbius band.

7 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 ; (2) s sp 1 p 2 and s 2; (3) 2 s sp + s 2 4p + s 2 4; (4) 2zp s 2 zs 1 for all z D.

8 Magic functions Define a rational function Φ z (s, p) of complex numbers z, s, p by Φ z (s, p) = 2zp 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.

9 A characterization of Γ-contractions For operators S, P let ρ(s, P ) = 1 2 [(2 S) (2 S) (2P S) (2P 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, ( ) 2αP S ( ) 2αP S 1 = 1 Φ α (S, P ) Φ α (S, P ) 0. 2 αs 2 αs

10 A sketch of sufficiency Suppose that ρ(αs, α 2 P ) 0 for all α D. polynomial g such that g 1 on G. Consider a 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 1 g(s, p)g(s, p) = ρ( ωs, T ω2 p) E(dω)F (ω, s, p), F (ω, s, p) for all (s, p) Γ. Apply to the commuting pair (S, P ); the right hand side is clearly positive. Thus g(s, P ) 1.

11 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 = P P 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.

12 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 operator 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?

13 Yes Theorem (Agler-Y, 1999, 2000) 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 S 0 0 S 0, P 0 0 P 0. For any polynomial f in two variables, f(s, P ) is the compression to H of f( S, P ). Thus ( S, P ) is a dilation of (S, P ).

14 Outline of the proof 1 The main Lemma If Γ is a spectral set for a commuting 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 ).

15 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(T2 ), on H. By Arveson s Extension Theorem and Stinespring s Theorem 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 = Ψ(z 1 + z 2 ), P def = Ψ(z 1 z 2 ) on K have the desired properties: ( S, P ) is a Γ-unitary dilation of (S, P ).

16 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.

17 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) (2ωP S) (2ωP S) 0 for all ω T. (S, P ) is called a Γ-co-isometry if (S, P ) is a Γ-isometry.

18 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 forward 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 multiplication by a bounded analytic B(E)-valued function on H 2 (E).

19 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.

20 A model Γ-isometry Let E be a separable Hilbert space, let A be an operator on E and let ψ(z) = A + A z for z D. ψ is an operator-valued bounded analytic function on D. The Toeplitz operator T ψ on the Hardy space HE 2 by is given (T ψ f)(z) = ψ(z)f(z) = (A + A z)f(z) for f H 2 E, z D. Let S = T ψ, P = T z on HE 2. operator. Thus P is the forward shift

21 A model Γ-isometry 2 Then P P = 1 and P S = T z T ψ = T z T A+A z = T za+a = T A+A z = S, S = T A+A z = sup θ = 2w(A). A + A e iθ = sup θ Hence S 2 if and only if w(a) 1. 2 Re ( e iθ/2 A ) Proposition The commuting pair (T A+A z, T z ), acting on H 2 (E), is a Γ-isometry if and only if w(a) 1. Moreover, every pure Γ-isometry is of this form.

22 A first model for Γ-contractions Let (S, P ) be a Γ-contraction on H. There exist a superspace K of H, a Γ-coisometry (S, P ) on K and an orthogonal decomposition K = K 1 K 2 such that K 1, K 2 reduce both S and P ; (S, P ) K 1 is a Γ-unitary; (S, P ) is the restriction to the common invariant subspace H of (S, P ); (S, P ) K 2 is unitarily equivalent to (T A +A z, T z ) acting on H 2 (E), for some Hilbert space E and some operator A on E satisfying w(a) 1.

23 References [1] J. Agler and N. J. Young, A commutant lifting theorem for a domain in C 2 and spectral interpolation, J. Functional Analysis 161 (1999) [2] J. Agler and N. J. Young, Operators having the symmetrized bidisc as a spectral set, Proc Edinburgh Math. Soc. 43 (2000) [3] J. Agler and N. J. Young, A model theory for Γ-contractions, J. Operator Theory 49 (2003)