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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.
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