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

A functional model for commuting pairs

• f 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

• perators 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

D2 |f|.

If (S, P) = π(A, B), then for any polynomial g and f = g ◦ π, g(S, P) = f(A, B) ≤ sup

D2 |f| = sup D2 |g ◦ π| = sup Γ

|g|. 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

• perators (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

|g|. 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 S2 − 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 (Tz1+z2, Tz1z2) of analytic Toeplitz operators on H2(D2), restricted to the subspace H2

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 C2. Γ is starlike about 0 but not circled. Γ ∩ R2 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 M¨

• bius band.

Characterizations of Γ

The following statements are equivalent for (s, p) ∈ C2. (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| + |s2 − 4p| + |s|2 ≤ 4; (4)

• 2zp − s

2 − zs

• ≤ 1

for all z ∈ D.

Magic functions

Define a rational function Φz(s, p) of complex numbers z, s, p by Φz(s, p) = 2zp − s 2 − zs . By the last slide, for any z ∈ D, Φz maps Γ into D−. Conversely, if (s, p) ∈ C2 is such that |Φz(s, p)| ≤ 1 for all z ∈ D then (s, p) ∈ Γ. This observation gives an analytic criterion for membership

• f Γ.

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, α2P) ≥ 0 for all α ∈ D. Necessity: for α ∈ D, Φα is analytic on a neighbourhood of Γ and |Φα| ≤ 1 on Γ. Hence, if (S, P) is a Γ-contraction, 1 −

2αP − S

2 − αS

∗ 2αP − S

2 − αS

• = 1 − Φα(S, P)∗Φα(S, P) ≥ 0.

A sketch of sufficiency

Suppose that ρ(αS, α2P) ≥ 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 1 − g(s, p)g(s, p) =

• T ρ(¯

ωs, ¯ ω2p) 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.

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 = U1 + U2 and P = U1U2 for some commuting pair of unitaries U1, U2 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

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 ∼

  

∗ ∗ S ∗ ∗ ∗

   ,

˜ P ∼

  

∗ ∗ P ∗ ∗ ∗

   .

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

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 P2 be the algebra of polynomials in two variables, and for f ∈ P2 let f♯ ∈ C(T2) be defined by f♯(z1, z2) = f(z1 + z2, z1z2). The map f → f♯ is an algebra-embedding of P2 in C(T2) 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(T2), 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(T2) → B(K) such that f(S, P) = θ(f♯) = PHΨ(f♯)|H for all polynomials f. The operators ˜ S def = Ψ(z1 + z2), ˜ P def = Ψ(z1z2)

• n 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 H2(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) − (2ωP − S)∗(2ωP − S) ≥ 0 for all ω ∈ T. (S, P) is called a Γ-co-isometry if (S∗, P ∗) is a Γ-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 H2(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 H2(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 = H1 ⊕ H2 such that (1) H1, H2 are reducing subspaces of both S and P, (2) (S, P)|H1 is a Γ-unitary, (3) (S, P)|H2 is a pure Γ-isometry.

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 Tψ on the Hardy space H2

E is given

by (Tψf)(z) = ψ(z)f(z) = (A + A∗z)f(z) for f ∈ H2

E , z ∈ D.

Let S = Tψ, P = Tz on H2

E .

Thus P is the forward shift

• perator.

A model Γ-isometry 2

Then P ∗P = 1 and P ∗S = T ∗

z Tψ = T¯ zTA+A∗z = T¯ zA+A∗ = T ∗ A+A∗z = S∗,

S = TA+A∗z = sup

θ

A + A∗eiθ = sup

θ

2 Re

• eiθ/2A∗
• = 2w(A).

Hence S ≤ 2 if and only if w(A) ≤ 1. Proposition The commuting pair (TA+A∗z, Tz), acting on H2(E), is a Γ-isometry if and only if w(A) ≤ 1. Moreover, every pure Γ-isometry is of this form.

A first model for Γ-contractions

Let (S, P) be a Γ-contraction on H. There exist a super- space K of H, a Γ-coisometry (S♭, P ♭) on K and an orthog-

• nal decomposition K = K1 ⊕ K2 such that
• K1, K2 reduce both S♭ and P ♭;
• (S♭, P ♭)|K1 is a Γ-unitary;
• (S, P) is the restriction to the common invariant sub-

space H of (S♭, P ♭);

• (S♭, P ♭)|K2 is unitarily equivalent to (TA∗+A¯

z, T¯ z) acting

• n H2(E), for some Hilbert space E and some operator A
• n E satisfying w(A) ≤ 1.

References

[1] J. Agler and N. J. Young, A commutant lifting theorem for a domain in C2 and spectral interpolation, J. Functional Analysis 161 (1999) 452–477. [2] J. Agler and N. J. Young, Operators having the sym- metrized bidisc as a spectral set, Proc Edinburgh Math.

• Soc. 43 (2000) 195-210.

[3] J. Agler and N. J. Young, A model theory for Γ-contractions,

• J. Operator Theory 49 (2003) 45-60.