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

Γdef= {(z + w, zw) : |z| ≤ 1, |w| ≤ 1}.

For any commuting pair (A,B) of contractions on a Hilbertspace 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 ofoperators 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 onthe 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)‖ ≤ supD2|f |.

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

‖g(S, P )‖ = ‖f(A,B)‖ ≤ supD2|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 linearoperators (on a Hilbert space H) for which the symmetrizedbidisc

Γdef= {(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 )‖ ≤ supΓ|g|.

If (S, P ) is a Γ-contraction then ‖S‖ ≤ 2 and ‖P‖ ≤ 1 (takeg to be a co-ordinate functional).

If A,B are commuting contractions then (A + B,AB) is aΓ-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 H2sym of symmetric func-

tions, is a Γ-contraction that is not of the form (A+B,AB).

Some properties of the symmetrized bidisc

Γdef= {(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

bΓdef= {(z + w, zw) : |z| = |w| = 1},

which is homeomorphic to the Mobius 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− s2− 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− s2− 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

of Γ.

A characterization of Γ-contractions

For operators S, P let

ρ(S, P ) = 12[(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 ifand only if

ρ(αS, α2P ) ≥ 0 for all α ∈ D.

Necessity: for α ∈ D, Φα is analytic on a neighbourhood ofΓ and |Φα| ≤ 1 on Γ. Hence, if (S, P ) is a Γ-contraction,

1−(

2αP − S2− αS

)∗ (2αP − S2− αS

)= 1−Φα(S, P )∗Φα(S, P ) ≥ 0.

A sketch of sufficiency

Suppose that ρ(αS, α2P ) ≥ 0 for all α ∈ D. Consider apolynomial g such that |g| ≤ 1 on G.

By Ando’s Theorem, ‖g(A+B,AB)‖ ≤ 1 for all commutingpairs (A,B).

Use this property to prove an integral representation formulafor 1 − g∗g. There exist a Hilbert space E, a B(E)-valuedspectral 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) =∫Tρ(ωs, ω2p) 〈E(dω)F (ω, s, p), F (ω, s, p)〉

for all (s, p) ∈ Γ. Apply to the commuting pair (S, P ); theright 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 ⊕ Ghaving 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 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 commutingpair (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, andfor f ∈ P2 let f ] ∈ C(T2) be defined by

f ](z1, z2) = f(z1 + z2, z1z2).

The map f 7→ 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 ) impliesthat θ is a completely contractive representation of thealgebra 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 ∗-representationΨ : C(T2)→ B(K) such that

f(S, P ) = θ(f ]) = PHΨ(f ])|H for all polynomials f.

The operators

Sdef= Ψ(z1 + z2), P

def= Ψ(z1z2) 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 suchthat 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-trivialreducing subspace of H on which V is unitary.

A pure isometry V is unitarily equivalent to multiplicationby 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 thefollowing 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 = kerP .

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 anda pure isometry (a forward shift operator) (Wold-Kolmogorov).

Every Γ-isometry is the orthogonal direct sum of a Γ-unitaryand a pure Γ-isometry. That is:

Let (S, P ) be a Γ-isometry on H. There exists an orthogonaldecomposition 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 H2E is given

by

(Tψf)(z) = ψ(z)f(z) = (A+A∗z)f(z) for f ∈ H2E , z ∈ D.

Let S = Tψ, P = Tz on H2E . Thus P is the forward shift

operator.

A model Γ-isometry 2

Then P ∗P = 1 and

P ∗S = T ∗z Tψ = TzTA+A∗z = TzA+A∗ = T ∗A+A∗z = S∗,

‖S‖ = ‖TA+A∗z‖ = supθ‖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-onal 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∗+Az, Tz) actingon H2(E), for some Hilbert space E and some operator Aon 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.