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
def
Γ = {(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 kSk ≤ 2 and kP k ≤ 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,
kf (A, B)k ≤ sup |f |.
D2
If (S, P ) = π(A, B), then for any polynomial g and f = g ◦ π,
kg(S, P )k = kf (A, B)k ≤ sup |f | = sup |g ◦ π| = sup |g|.
D2
D2
That is, Γ is a spectral set for the pair (S, P ).
Γ
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
def
Γ = {(z + w, zw) : |z| ≤ 1, |w| ≤ 1}
is a spectral set.
This means that, for all scalar polynomials g in two variables,
kg(S, P )k ≤ sup |g|.
Γ
If (S, P ) is a Γ-contraction then kSk ≤ 2 and kP k ≤ 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 − 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
2
H 2(D2), restricted to the subspace Hsym
of symmetric functions, is a Γ-contraction that is not of the form (A + B, AB).
Some properties of the symmetrized bidisc
def
Γ = {(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
def
bΓ = {(z + w, zw) : |z| = |w| = 1},
which is homeomorphic to the Möbius 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 − s̄p| ≤ 1 − |p|2 and |s| ≤ 2;
(3)
2|s − s̄p| + |s2 − 4p| + |s|2 ≤ 4;
(4)
2zp − s ≤1
2 − zs
for all z ∈ D.
Magic functions
Define a rational function Φz (s, p) of complex numbers z, s, p
by
2zp − s
Φz (s, p) =
.
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
of Γ.
A characterization of Γ-contractions
For operators S, P let
∗ (2 − S) − (2P − S)∗ (2P − S)]
ρ(S, P ) = 1
[(2
−
S)
2
= 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,
2αP − S ∗ 2αP − S
1−
= 1 − Φα(S, P )∗Φα(S, P ) ≥ 0.
2 − αS
2 − αS
A sketch of sufficiency
Suppose that ρ(αS, α2P ) ≥ 0 for all α ∈ D.
polynomial g such that |g| ≤ 1 on G.
Consider a
By Ando’s Theorem, kg(A + B, AB)k ≤ 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) =
Z
ρ(ω̄s, ω̄ 2p) hE(dω)F (ω, s, p), F (ω, s, p)i
T
for all (s, p) ∈ Γ. Apply to the commuting pair (S, P ); the
right hand side is clearly positive. Thus kg(S, P )k ≤ 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 = P P ∗ and P ∗S = S ∗ and kSk ≤ 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 P̃ on a Hilbert space
K ⊃ H.
By the Commutant Lifting Theorem, there exists an operator S̃ on K which commutes with P̃ , has norm kSk and is
a dilation of S.
It does not follow that (S̃, P̃ ) is a Γ-unitary, or even a Γcontraction.
Can we choose S̃ so that (S̃, P̃ ) 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 (S̃, P̃ ) on G∗ ⊕ H ⊕ G
having block operator matrices of the forms

∗ 0 0



S̃ ∼ ∗ S 0 ,
∗ ∗ ∗


∗ 0 0

0 .
∗ ∗ ∗

P̃ ∼ ∗ P
For any polynomial f in two variables, f (S, P ) is the compression to H of f (S̃, P̃ ). Thus (S̃, P̃ ) 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 7→ f ] is an algebra-embedding of P2 in C(T2)
]
Let its range be P2.
]
Define an algebra representation θ : P2 → 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 P2 ⊂ 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(T2) → B(K) such that
f (S, P ) = θ(f ]) = PHΨ(f ])|H
for all polynomials f.
The operators
def
def
S̃ = Ψ(z1 + z2), P̃ = Ψ(z1z2)
on K
have the desired properties: (S̃, P̃ ) is a Γ-unitary dilation of
(S, P ).
Isometries
For V ∈ B(H), the following statements are equivalent:
(1) kV xk = kxk 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
(S̃, P̃ ) to a joint invariant subspace of (S̃, P̃ ).
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 kSk ≤ 2;
(3) kSk ≤ 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 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).
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 HE2 is given
by
(Tψ f )(z) = ψ(z)f (z) = (A + A∗z)f (z)
for f ∈ HE2, z ∈ D.
Let S = Tψ , P = Tz on HE2. Thus P is the forward shift
operator.
A model Γ-isometry 2
Then P ∗P = 1 and
∗
∗
P ∗S = Tz∗Tψ = Tz̄ TA+A∗z = Tz̄A+A∗ = TA+A
∗z = S ,
∗
iθ
iθ/2
∗
kSk = kTA+A∗z k = sup kA + A e k = sup k2 Re e
A k
θ
θ
= 2w(A).
Hence kSk ≤ 2 if and only if w(A) ≤ 1.
Proposition The commuting pair (TA+A∗z , Tz ), 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.
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 = 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 subspace H of (S [, P [);
• (S [, P [)|K2 is unitarily equivalent to (TA∗+Az̄ , Tz̄ ) acting
on H 2(E), for some Hilbert space E and some operator A
on 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 symmetrized 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.