A functional model for commuting pairs of contractions and the symmetrized bidisc Nicholas Young Leeds and Newcastle Universities Lecture 3 A functional model for commuting pairs of contractions St Petersburg, June 2016
Γ-contractions recapitulation A commuting pair (S, P ) of operators on a Hilbert space H is a Γ-contraction if, for every polynomial g in two variables, g(s, P ) sup g. Γ If A, B are commuting contractions then (A + B, AB) is a Γ-contraction. If (S, P ) is a commuting pair of operators then (S, P ) is a Γ-contraction if and only if ρ(αs, α 2 P ) 0 for all α D, where ρ(s, P ) = 2(1 P P ) S + S P S + P S. By continuity for any Γ-contraction (S, P ) and any ω T, 2(1 P P ) ω(s S P ) ω(s P S) 0.
The fundamental operator of a Γ-contraction Let (S, P ) be a Γ-contraction on H. conjugate by DP 1 obtain If P < 1, we may = (1 P P ) 1 2 in the last inequality to 1 Re{ωDP 1 (S S P )DP 1 } 0 for all ω T. Hence the operator F = DP 1 (S S P )DP 1 B(D P ) satisfies w(f ) 1. For a general Γ-contraction (S, P ) there exists a unique F B(D P ) such that S S P = D P F D P, and this unique F satisfies w(f ) 1. This F is called the fundamental operator of the Γ-contraction (S, P ) [4].
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 contraction). Let F be the fundamental operator of (S, P ). Then (S, P ) is unitarily equivalent to the pair (S, P) on the Hilbert space H P def = H 2 (D P ) Θ P H 2 (D P ) where (S, P) is the compression of the Γ-isometry ( T F +zf, T z ) on H 2 (D P ) to its co-invariant subspace H P. Observe that P is precisely the Nagy-Foias model of P.
Proof Recall the proof of the Nagy-Foias model. We use Θ P (λ) = [ P + λd P (1 λp ) 1 D P ] D P. The map given by U : H H P def = H 2 (D P ) Θ P H 2 (D P ) is a unitary operator, and (Ux)(z) = D P (1 H zp ) 1 x UP U = the compression to H P of T z 1 DP = P.
Proof 2 Since F is the fundamental operator of (S, P ), we have F B(D P ), w(f ) 1, and S SP = D P F D P. Since w(f ) 1, the pair ( T F +zf, T z ) is a Γ-isometry on H 2 (D P ). We shall next prove that W T F +zf = SW. Observe that W : H 2 (D P ) H satisfies W (z n y), x = z n y, DP x + zd P P x +... = y, D P P n x = P n D P y, x. Thus W (z n y) = P n D P y.
Proof 3 Hence W T F +zf (z n y) = W (z n F y + z n+1 F y) = P n D P F y + P n+1 D P F y, while SW z n y = SP n D P y = P n SD P y. Therefore (W T F +zf SW )(z n y) = P n Hy, where H = D P F + P D P F SD P : DP H.
Proof 4 H = D P F + P D P F SD P : D P H. Now HD P = D P F D P + P D P F D P SDP 2 = (S P S ) + P (S SP ) S(1 P P ) = 0. Hence H = 0, and so W T F +zf = SW. Accordingly (T F +zf ) W = W S.
Proof 5 The equation (T F +zf ) W = W S implies that (T F +zf ) H P = T F +zf ran W ran W = H P. Thus H P is a co-invariant subspace of H 2 (D P ) for T F +zf. In the equation (T F +zf ) W = W S : H H 2 (D P ) take the range restriction U of W to obtain (T F +zf ) U = US : H H P.
Proof 6 Hence (T F +zf ) H P = US U : H P H P. Finally USU = the compression of T F +zf to H P = S. We have shown that U(S, P )U = (S, P).
Completely non-unitary Γ-contractions Every Γ-contraction is the orthogonal direct sum of a Γ- unitary and a completely non-unitary Γ-contraction, meaning one for which the second component is a completely non-unitary contraction. Let (S, P ) be a Γ-contraction on H. Let H 1 be the maximal subspace of H on which P is unitary and let H 2 = H H 1. Then (1) H 1, H 2 reduce both S and P (2) (S H 1, P H 1 ) is Γ-unitary (3) (S H 2, P H 2 ) is a Γ-contraction and P H 2 is completely non-unitary.
A functional model for Γ-contractions Theorem (Sarkar 2015) Let (S, P ) be a completely non-unitary Γ-contraction. Let H P be the Nagy-Foias model space of the contraction P : H P = [ H 2 (D P ) P L 2 (D P ) ] { Θ P u P u : u H 2 (D P ) }. Then (S, P ) is unitarily equivalent to (S, P), where P is the Nagy-Foias model of P and S is the compression to H P of the operator T F +zf U on H 2 (D P ) P L 2 (D P ), where F is the fundamental operator of (S, P ) and U B( P L 2 (D P )) is such that (U, M e it) is Γ-unitary on P L 2 (D P ).
References [1] B. Sz.-Nagy and C. Foias, Harmonic Analysis of operators on Hilbert space, Akadémiai Kiadő, Budapest 1970. [2] J. Agler and N. J. Young, A model theory for Γ-contractions, J. Operator Theory 49 (2003) 45-60. [3] T. Bhattacharyya and S. Pal, A functional model for pure Γ-contractions, J. Operator Theory 71 (2014). [4] T. Bhattacharyya, S. Pal and S. Shyam Roy, Dilation of Γ-contractions by solving operator equations, Advances in Mathematics 230 (2012) 577-606. [5] J. Sarkar, Operator theory on symmetrized bidisc, Indiana Univ. Math. J. 64 (2015) 847 873.
More about the symmetrized bidisc The domain G def = {(z + w, zw) : z < 1, w < 1}, the interior of Γ, is called the open symmetrized bidisc. There are at least four reasons to take an interest in G, connected to classical complex function theory of the unit disc D the Nagy-Foias theory of models of operators the theory of invariant distances in several complex variables a problem in robust control theory.
The 2 2 spectral Nevanlinna-Pick problem Given distinct points λ 1,..., λ N D and square matrices W 1,..., W N C 2 2, none of them a scalar multiple of the identity, ascertain whether there exists an analytic function F : D C 2 2 such that F (λ j ) = W j for j = 1,..., N and r(f (λ)) 1 for all λ D where r( ) is the spectral radius. This problem is equivalent to an interpolation problem for analytic functions f : D G. This is because r(w ) 1 if and only if (tr W, det W ) Γ.
A Schwarz-Pick Lemma for the spectral radius Let Ω be the set of 2 2 matrices of spectral radius < 1. Let W 1, W 2 Ω be non-scalar matrices and let λ 1, λ 2 D. Let s j = tr W j, p j = det W j for j = 1, 2. There exists an analytic function F F (λ j ) = W j for j = 1, 2 if and only if sup ω =1 (s 2 p 1 s 1 p 2 )ω 2 + 2(p 2 p 1 )ω + s 1 s 2 (s 1 s 2 p 1 )ω 2 2(1 p 2 p 1 )ω + s 2 p 2 s 1 : D Ω such that λ 1 λ 2 1 λ 2 λ 1. This result comes out of the study of the Carathéodory and Kobayashi distances on G.
The Γ-Nevanlinna-Pick problem Given distinct points λ 1,..., λ N D and points (s 1, p 1 ),..., (s N, p N ) in Γ, determine whether there exists an analytic function h : D Γ such that h(λ j ) = (s j, p j ) for j = 1,..., N. Is there a Pick-type criterion for solvability? Answer: almost certainly not. We can solve the Γ-Nevanlinna-Pick problem in the case that N = 2.
Weak solvability Say that a Γ-Nevanlinna-Pick problem λ j D (s j, p j ) G for j = 1,..., N (1) is weakly solvable if, for every analytic map g : G D, the (classical) Nevanlinna-Pick problem λ j g(s j, p j ) is solvable. Clearly solvable problems are weakly solvable (if h satisfies conditions (1) then g h maps λ j to g(s j, p j )). Theorem. The Γ-Nevanlinna-Pick problem (1) is weakly solvable if and only if, for all ω T, [ ] (2 ω si )(2 ωs j ) (2 ω p i s i )(2ωp j s j ) N 1 λ i λ j i,j=1 0.
Two-point Γ-Nevanlinna-Pick problems Theorem. Let λ 1, λ 2 be distinct points in D and let (s 1, p 1 ), (s 2, p 2 ) belong to G. The Γ-Nevanlinna-Pick problem λ j D (s j, p j ) for j = 1, 2 (2) is solvable if and only if it is weakly solvable. Corollary. The Γ-Nevanlinna-Pick problem (2) is solvable if and only if, for all ω T, [ ] (2 ω si )(2 ωs j ) (2 ω p i s i )(2ωp j s j ) 2 1 λ i λ j i,j=1 0. For the three-point Γ-Nevanlinna-Pick problem, weak solvability does not imply solvability.
The Carathéodory distance Let d denote the hyperbolic distance on D: for λ, µ D, d(λ, µ) def = tanh 1 λ µ 1 µλ. For any bounded domain Ω C n, the Carathéodory distance on Ω is defined for any points z, w Ω by C Ω (z, w) def = sup{d(g(z), g(w)) : g is analytic from Ω to D}. Any analytic function g : Ω D for which the supremum is attained is called a Carathéodory extremal function for z, w in Ω.
The Carathéodory distance on G The above solvability criterion for two-point Γ-Nevanlinna- Pick problems translates into the following statement. Theorem For any pair of points z 1 = (s 1, p 1 ), z 2 = (s 2, p 2 ) in G, there is a Carathéodory extremal function of the form Φ ω for some ω T, and therefore C G (z 1, z 2 ) = sup d(φ ω (z 1 ), Φ ω (z 2 )) ω =1 = tanh 1 sup ω =1 (s 2 p 1 s 1 p 2 )ω 2 + 2(p 2 p 1 )ω + s 1 s 2 (s 1 s 2 p 1 )ω 2 2(1 p 2 p 1 )ω + s 2 p 2 s 1.
The Kobayashi distance For any bounded domain Ω C n, the Lempert function on Ω is defined for any points z, w Ω to be δ Ω (z, w) def = inf d(λ, µ) over all λ, µ D such that there exists an analytic map f : D Ω such that f(λ) = z and f(µ) = w. δ Ω does not in general satisfy the triangle inequality. The Kobayashi distance K Ω on Ω is defined to be the greatest distance K on Ω such that K δ Ω. Since C Ω δ Ω, always C Ω K Ω.
The Kobayashi distance on G Lempert s theorem states that, for any bounded convex domain Ω, C Ω = K Ω = δ Ω. C. Costara proved that G is not isomorphic to any convex domain. Nevertheless: Theorem. C G = K G = δ G. This theorem solved a long-standing problem in the theory of invariant distances: are all domains for which C = K isomorphic to convex domains?
References J. Agler and N. J. Young, The hyperbolic geometry of the symmetrised bidisc, J. Geometric Analysis 14 (2004) 375-403. N. J. Young, Some analysable instances of mu-synthesis, Mathematical methods in systems, optimization and control, Editors: H. Dym, M. de Oliveira, M. Putinar, Operator Theory: Advances and Applications 222 349 366, Birkhäuser, Basel, 2012.