A model for commuting pairs of contractions on Hilbert space - - PowerPoint PPT Presentation

A model for commuting pairs of contractions on Hilbert space

Nicholas Young

Leeds and Newcastle Universities Newcastle, March 2017

Synopsis

• Contractions and shifts
• The Nagy-Foias functional model of a contraction
• The symmetrized bidisc Γ – an interesting set in C2
• Γ-analogues of contractions, unitaries and isometries
• A Nagy-Foias functional model for Γ-contractions

The idea of a model of an operator

There is no ‘canonical form’ of bounded linear operators on Hilbert space under unitary equivalence. The next best thing would be to express the general oper- ator as a part of a well-understood operator having a rich structure. This is the purpose of the book Harmonic analysis of operators on Hilbert space, by C. Foias and B. Sz.-Nagy, 1966. It expresses a general operator, up to unitary equivalence, in terms of ‘shift operators’ on Hilbert spaces of functions having analytic structure.

Contractions

A contraction is a linear operator P on a Hilbert space H such that P ≤ 1. If P is a bounded linear operator on H then P is a contraction if idH − P ∗P ≥ 0. The defect operator DP of a contraction P on H is the positive operator (1 − P ∗P)

1 2, acting on H.

The defect operator of P ∗ is thus DP ∗ = (1 − PP ∗)

1 2.

The defect space DP of P is ran DP, a subspace of H. We have PDP ⊂ DP ∗.

Some basic objects

Let E be a separable Hilbert space. L2(E) is the Lebesgue space of square-integrable E-valued functions f on T with norm f2 =

• T |f(z)|2m(dz)

where m is normalized Lebesgue measure on T. H2(E) is the Hardy space of analytic E-valued functions on D. H2(E) can be thought of as the closed subspace of L2(E) comprising the functions f whose negative Fourier coeffi- cients ˆ f(n), n < 0, vanish.

The shift operator

The shift operator on H2(E) is the operator Tz defined for f ∈ H2(E) by (Tzf)(z) = zf(z) for z ∈ T. Note that Tz shifts the Taylor coefficients of a function in H2(E): Tz(a0 + a1z + a2z2 + . . . ) = 0 + a0z + a1z2 + . . . . The adjoint operator T ∗

z on H2(E) shifts coefficients back-

wards: T ∗

z (a0 + a1z + a2z2 + . . . ) = a1 + a2z + a3z2 + . . . .

Tz and T ∗

z are well understood operators.

The Nagy-Foias functional model of a contraction

An embedding of H in H2(DP ∗)

For a contraction P on H, define W : H → H2(DP ∗) by Wx ∼ (DP ∗x, DP ∗P ∗x, DP ∗(P ∗)2x, . . . ), Wx(z) = DP ∗(1 − zP ∗)−1x. For any x ∈ H, Wx2 = x2 − lim

N→∞ (P ∗)Nx2.

Hence, if P is a ‘pure contraction’, meaning that (P ∗)N tends strongly to zero on H as N → ∞, then W is an iso- metric embedding of H in H2(DP ∗). Moreover WP ∗x = (DP ∗P ∗x, DP ∗(P ∗)2x, . . . ) = T ∗

z (DP ∗x, DP ∗P ∗x, . . . )

= T ∗

z Wx

where Tz denotes the shift operator on H2(DP ∗).

A functional model - part 1

Suppose P is a pure contraction. Let E denote the range of W. E is an invariant subspace of H2(DP ∗) with respect to T ∗

z .

Let U : H → E be the range restriction of W. Thus U is a unitary operator, and P = U∗(T ∗

z |E)∗U = U∗( the compression of Tz to E)U.

We need an effective description of the space E = ran W in terms of the operator P.

The characteristic operator function ΘP

This is the analytic operator-valued function on D given by ΘP(λ) = [−P + λDP ∗(1 − λP ∗)−1DP]|DP for λ ∈ D. Its values are contractive operators from DP to DP ∗, and ran W = H2(DP ∗) ⊖ ΘPH2(DP). The functional model for pure contractions If P is a pure contraction then P is unitarily equivalent to the compression of the shift operator Tz on H2(DP ∗) to its co-invariant subspace H2(DP ∗) ⊖ ΘPH2(DP).

Completely non-unitary contractions

Consider a contraction P on a Hilbert space H. If P has a nonzero invariant subspace H1 on which it is isometric then P splits up into a block operator P =

• P1

P2

• with repect to the orthogonal decomposition H1 ⊕ H⊥

1 of

• H. Here P1 is a unitary operator, and so, by the Spectral

Theorem, P1 =

• T λE(dλ)

for some spectral measure E. P2 is a c.n.u. contraction: it has no nonzero unitary restric- tion.

The model space HP

Let P be a c.n.u contraction on H. Define an operator- valued function on T by ∆P(eit) = [1 − ΘP(eit)∗ΘP(eit)]

1 2.

For almost all t ∈ R, ∆P(eit) is an operator on DP. The model space HP is defined by HP =

• H2(DP ∗) ⊕ ∆PL2(DP)
• ⊖
• ΘPu ⊕ ∆Pu : u ∈ H2(DP)
• .

HP is a space of functions on T with values in DP ∗ ⊕ DP.

Theorem: the Nagy-Foias functional model

Let P be a c.n.u contraction on H. Then P is unitarily equivalent to the operator P on the model space HP =

• H2(DP ∗) ⊕ ∆PL2(DP)
• ⊖
• ΘPu ⊕ ∆Pu : u ∈ H2(DP)
• given by

P∗(u ⊕ v) = e−it[u(eit) − u(0)] ⊕ e−itv(eit)

for all u ⊕ v ∈ HP.

P is the Nagy-Foias model of P.

Commuting pairs of contractions

Let A and B be c.n.u contractions on a Hilbert space H such that AB = BA. A and B have Nagy-Foias models, but they typically act on different Hilbert spaces. No canonical model is known for the pair (A, B). Instead, we exhibit a canonical model for the symmetrization

• f (A, B), meaning the pair of operators (A + B, AB).

(A + B, AB) is called a Γ-contraction.

The symmetrized bidisc: an interesting set in C2

The symmetrized bidisc

The closed symmetrized bidisc is the set 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.

Magic functions

Define a rational function Φz(s, p) of complex numbers z, s, p by Φz(s, p) = 2zp − s 2 − zs . 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.

Γ-analogues of contractions, unitaries and isometries

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 Ando’s inequality.

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).

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.

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.

Γ-unitary dilations

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).

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 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 analytic 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. Then (S, P) has an extension to the orthogonal direct sum

• f
• a Γ-unitary and
• a Γ-coisometry (TA∗+A¯

z, T¯ z) acting on H2(E),

for some Hilbert space E and some operator A on E satisfying w(A) ≤ 1.

A Nagy-Foias functional model for Γ-contractions

History

• J. Agler and N. J. Young, A model theory for Γ-contractions,
• J. Operator Theory 49 (2003) 45-60.
• T. Bhattacharyya and S. Pal, A functional model for pure

Γ-contractions, J. Operator Theory 71 (2014).

• J. Sarkar, Operator theory on symmetrized bidisc, Indiana
• Univ. Math. J.

64 (2015) 847–873.

The fundamental operator of a Γ-contraction

Let (S, P) be a Γ-contraction on H. If P < 1, we may conjugate by D−1

P

= (1−P ∗P)−1

2 in the inequality ρ(S, P) ≥ 0

to obtain 1 − Re{ωD−1

P (S − S∗P)D−1 P } ≥ 0

for all ω ∈ T. Hence the operator F = D−1

P (S −S∗P)D−1 P

∈ B(DP) satisfies w(F) ≤ 1. For a general Γ-contraction (S, P) there exists a unique F ∈ B(DP) such that S − S∗P = DPFDP, and this unique F satisfies w(F) ≤ 1. This F is called the fundamental operator

• f the Γ-contraction (S, P).

A functional model for pure Γ-contractions

Theorem (Bhattacharyya and Pal, 2014) Let (S, P) be a pure Γ-contraction (so that P is a pure con- traction). Let F∗ be the fundamental operator of (S∗, P ∗). Then (S, P) is unitarily equivalent to the pair (S, P) on the Hilbert space

HP

def

= H2(DP ∗) ⊖ ΘPH2(DP) where (S, P) is the compression of the Γ-isometry

• TF ∗

∗ +zF∗, Tz

• n H2(DP ∗) to its co-invariant subspace HP.

Observe that P is precisely the Nagy-Foias model of P.

A functional model for Γ-contractions

Theorem (Sarkar 2015) Let (S, P) be a completely non-unitary Γ-contraction. Let HP be the Nagy-Foias model space of the contraction P: HP =

• H2(DP ∗) ⊕ ∆PL2(DP)
• ⊖
• ΘPu ⊕ ∆Pu : u ∈ H2(DP)
• .

Then (S, P) is unitarily equivalent to (S, P), where P is the Nagy-Foias model of P and S is the compression to HP of the operator TF ∗

∗ +zF∗ ⊕ U on H2(DP ∗) ⊕ ∆PL2(DP),

where F∗ is the fundamental operator of (S∗, P ∗) and U ∈ B(∆PL2(DP)) is such that (U, Meit) is Γ-unitary on ∆PL2(DP).

A question

Suppose given a commuting pair of contractions (A, B). Is it possible to augment the canonical model of (A+B, AB) by some object which recovers the information lost by sym- metrization? Note that the symmetrization of a nonzero pair (A, B) can be (0, 0), for example A = −B =

• I
• .

Thank you!