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 1

The Nagy-Foias model of a contractive

• perator

St Petersburg, June 2016

Synopsis

The Nagy-Foias functional model expresses the general com- pletely non-unitary contractive linear operator T on Hilbert space as a compression of the shift operator on an L2 space

• f vector-valued analytic functions on the unit circle. The

compression is to a semi-invariant subspace HT, which has a concrete description in terms of the characteristic operator function

• f the operator.

The purpose of these lectures is to extend the Nagy-Foias functional model to pairs of commuting contractions, and indeed to a larger class, called Γ-contractions. To do this we need to study the symmetrized bidisc, which is the set Γ = {(z + w, zw) : |z| ≤ 1, |w| ≤ 1}.

Some basic objects

Throughout this lecture T is a bounded linear operator of norm at most one on a separable Hilbert space H. Such an

• bject is a contraction.

For any separable Hilbert space E, we denote by H2(E) the space of analytic E-valued functions on D of the form f(z) =

∞

• n=0

anzn, z ∈ D, with an ∈ E and f < ∞, where f2 =

∞

• n=0

an2

E.

L2(E) denotes the Lebesgue space of square-integrable E- valued functions on the circle.

The shift operator

The shift operator Tz

• n H2(E) is defined by

Tzf(z) = zf(z), z ∈ D,

• r equivalently

Tz(a0, a1, . . . ) = (0, a0, a1, . . . ). Its adjoint T ∗

z , the backward shift, is given by

T ∗

z f(z) = f(z) − f(0)

z

• r equivalently

T ∗

z (a0, a1, . . . ) = (a1, a2, . . . ).

To what extent do Tz and T ∗

z provide models for the general

Hilbert space contraction?

Defect operators and spaces

The defect operator DT of T is defined to be the positive

• perator (1 − T ∗T)

1 2, acting on H.

The defect operator of T ∗ is thus DT ∗ = (1 − TT ∗)

1 2.

Note that TDT = T(1 − T ∗T)

1 2 = (1 − TT ∗) 1 2T = DT ∗T.

The defect space DT of T is ran DT ⊂ H. We have TDT ⊂ DT ∗.

An embedding of H in H2(DT ∗)

Define a map W : H → H2(DT ∗) by Wx ∼ (DT ∗x, DT ∗T ∗x, DT ∗(T ∗)2x, . . . ), Wx(z) = DT ∗(1 − zT ∗)−1x. Then, for any x ∈ H, Wx2 = DT ∗x2 + DT ∗T ∗x2 + . . . = DT ∗x, DT ∗x + DT ∗T ∗x, DT ∗T ∗x + . . . = (1 − TT ∗)x, x + (1 − TT ∗)T ∗x, T ∗x + . . . = x2 − T ∗x2 + T ∗x2 − (T ∗)2x2 + (T ∗)2x2 − . . . = lim

N→∞ x2 − (T ∗)Nx2.

Hence, if (T ∗)N tends strongly to zero on H as N → ∞, then W is an isometric embedding of H in H2(DT ∗).

An intertwining

Say that T is a pure contraction if (T ∗)N → 0 as N → ∞. Let W : H → H2(DT ∗) be the isometric embedding just

• constructed. Then, for x ∈ H,

WT ∗x = (DT ∗T ∗x, DT ∗(T ∗)2x, . . . ) = T ∗

z (DT ∗x, DT ∗T ∗x, . . . )

= T ∗

z Wx

where Tz denotes the shift operator on H2(DT ∗). Let E denote the range of W. Observe that E is an invariant subspace of H2(DT ∗) with respect to T ∗

z .

Theorem

If T is a pure contraction then there is a unitary map U from H to an T ∗

z -invariant subspace E of H2(DT ∗) such that

T ∗ = U∗(T ∗

z |E)U.

Thus T = U∗(T ∗

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

We have proved this theorem: just let U : H → E be given by Ux = Wx. This is already a ‘functional model’ of sorts, at least for pure contractions. The shift operator on H2(DT ∗) can be analysed by means of classical function theory, whence the title of Nagy and Foias’ book, Harmonic analysis of operators on Hilbert space. To give the model power, we need an effective description

• f the space E = ran W in terms of the operator T.

The characteristic operator function ΘT

This is the operator-valued function on D given by ΘT(λ) = [−T + λDT ∗(1 − λT ∗)−1DT]|DT for λ ∈ D. Its values are contractive operators from DT to DT ∗. ΘT is a purely contractive analytic function,which means that:

• ΘT is analytic on D,
• its values are contractive operators, and
• for every vector x ∈ DT, ΘT(0)x < x.

Exercise: for z, w ∈ D, 1 − ΘT(z)ΘT(w)∗ = (1 − ¯ wz)DT ∗(1 − zT ∗)−1(1 − ¯ wT)−1DT ∗.

An identity

If MΘT : H2(DT) → H2(DT ∗) is the operation of pointwise multiplication by ΘT then WW ∗ + MΘT M∗

ΘT = 1H2(DT∗).

• Proof. Let k be the Szeg˝
• kernel: kw(z) = (1 − ¯

wz)−1 for z, w ∈ D. For any w ∈ D, x ∈ H and y ∈ DT ∗, W ∗(kw ⊗ y), xH = kw ⊗ y, WxH2(DT∗) =

• kw ⊗ y, DT ∗(1 − zT ∗)−1x
• H2(DT∗)

=

• y, DT ∗(1 − wT ∗)−1x
• DT∗

=

• (1 − ¯

wT)−1DT ∗y, x

• DT∗ .

Hence W ∗(kw ⊗ y) = (1 − ¯ wT)−1DT ∗y.

Proof of the identity continued

Hence

WW ∗(kw ⊗ y) (z) =

• W(1 − ¯

wT)−1DT ∗y

• (z)

= DT ∗(1 − zT ∗)−1(1 − ¯ wT)−1DT ∗y = (1 − ¯ wz)−1 1 − ΘT(z)ΘT(w)∗ . Now [MΘT M∗

ΘT (kw ⊗ y)](z) = ΘT(z)[kw ⊗ ΘT(w)∗y](z)

= kw(z)ΘT(z)ΘT(w)∗y. On adding these two equations we find [WW ∗ + MΘT M∗

ΘT ](kw ⊗ y) = kw ⊗ y,

and so WW ∗ + MΘT M∗

ΘT = 1H2(DT∗).

The range of W : H → H2(DT ∗)

• Proposition. If T is a pure contraction then

ran W = H2(DT ∗) ⊖ ΘTH2(DT) where ΘTH2(DT) def = {ΘTf : f ∈ H2(DT)} ⊂ H2(DT ∗). Proof. Since W is an isometry, WW ∗ is the orthogonal projection onto ran W. Hence MΘT M∗

ΘT = 1H2(DT∗) − WW ∗ is the projection on

(ran W)⊥. Therefore ran W = (ΘTH2)⊥.

Theorem - the functional model for pure contractions

If T is a pure contraction then T is unitarily equivalent to the compression of the shift operator Tz on H2(DT ∗) to its co-invariant subspace H2(DT ∗) ⊖ ΘTH2(DT).

Completely non-unitary contractions

A c.n.u. contraction is a contraction which has no nontrivial unitary restriction. If T is a contraction on H then the set K = {x ∈ H : T nx = x = (T ∗)n for n ≥ 1} is a T-reducing subspace of H on which T acts as a unitary

• perator. It is clearly the largest such subspace. Thus T is

the orthogonal direct sum of a unitary operator T|K and a c.n.u. contraction T|K⊥. Pure contractions are c.n.u. The functional model of the last slide can be extended to c.n.u. contractions.

The model space HT

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

1 2.

For almost all t ∈ R, ∆T(eit) is an operator on DT. The model space HT is defined by HT =

• H2(DT ∗) ⊕ ∆TL2(DT)
• ⊖
• ΘTu ⊕ ∆Tu : u ∈ H2(DT)
• .

HT is a space of functions on T with values in DT ∗ ⊕ DT.

Theorem: the Nagy-Foias functional model

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

• H2(DT ∗) ⊕ ∆TL2(DT)
• ⊖
• ΘTu ⊕ ∆Tu : u ∈ H2(DT)
• given by

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

for all u ⊕ v ∈ HT. Here ΘT(λ) =

• −T + λDT ∗(1 − T ∗λ)−1DT
• |DT.

T is the Nagy-Foias model of T. It is canonical.

Commuting pairs of contractions

Let T1, T2 be commuting contractions on a Hilbert space H Is there a canonical functional model of (T1, T2), of Nagy- Foias type? Claim: There is a canonical functional model of a Γ-contraction, (to be defined). This model can be interpreted as a model for a commuting pair of contractions. The first step is to construct a ‘canonical Γ-unitary dilation’.

Unitary dilations

Let T be an operator on a Hilbert space H. Consider an

• perator V on the space G∗ ⊕H⊕G (for some Hilbert spaces

G∗, G) of the form V ∼

  

∗ ∗ T ∗ ∗ ∗

   .

For any polynomial f, the compression of f(V ) to H is f(T). An operator V (on a superspace of H) with this property is called a dilation

• f T.

Theorem (B. Sz.-Nagy, 1953) Every contraction on a Hilbert space has a unitary dilation. The minimal unitary dilation of a contraction is unique up to unitary equivalence.

Ando’s theorem

A commuting pair T1, T2 of commuting contractions on a Hilbert space H has a simultaneous unitary dilation. That is, there exists a superspace K of H and a commuting pair U1, U2 of unitaries on K such that, for every polynomial f in two variables, f(T1, T2) is the compression to H of f(U1, U2). Equivalently, for some orthogonal decomposition G∗ ⊕ H ⊕ G

• f K, and for j = 1, 2,

Uj ∼

  

∗ ∗ Tj ∗ ∗ ∗

   .

A drawback to Ando’s theorem

The simultaneous unitary dilation provided by Ando’s theo- rem is not canonical. That is, not all such dilations (U1, U2) of a given pair (T1, T2) (even minimal) are unitarily equivalent. This fact suggests that there might not be a canonical func- tional model for commuting pairs of contractions. In the next lecture we shall see a slight modification of viewpoint which circumvents this obstacle. We shall introduce the notion of a Γ-contraction, where Γ is the set Γ = {(z + w, zw) : |z| ≤ 1, |w| ≤ 1}.

References

[1] B. Sz.-Nagy and C. Foias, Harmonic Analysis of opera- tors on Hilbert space, Akad´ emiai Kiad˝

• , Budapest 1970.

[2] N. K. Nikolskii and V. I. Vasyunin, Elements of spec- tral theory in terms of the free function model, Part I: Ba- sic constructions, in Holomorphic Spaces, ed. S. Axler, J.

• E. McCarthy and D. Sarason, MSRI Publications, Berkeley,

1998, pages 211-302.