Contractive determinantal representations of stable polynomials on a matrix polyball

Transcription

1 Contractive determinantal representations of stable polynomials on a matrix polyball Anatolii Grinshpan, Dmitry S. Kaliuzhnyi-Verbovetskyi, Victor Vinnikov, and Hugo J. Woerdeman Drexel University, Philadelphia, USA June, 2015

2 Determinantal representation We study determinantal representations of polynomials k p(z) = p(0) det(i KZ ), Z = Z r I nr, (1) r=1 z = (z (r) ) r=1,...,k;i=1,...,lr,j=1,...,m r, Z r = (z (r) lr mr ) i=1,j=1. Thus d = k r=1 l r m r variables. Denote B = {z : Z < 1}. QUESTION: Can we choose K 1 as soon as p(z) 0 on B (=p stable)? For ( z (1) 11 z (1) 12 ) < 1 and z (2) 11 < 1, 0 1 z(1) z(1) z(2) = det z(1) 11 z(2) 11 z(1) 12 z(2) ( 50 z (1) 11 z (1) z (2) 11 ).

3 Von Neumann Inequality, 1951 Given an analytic function f (z) = i 0 f iz i, z < 1, and a strict contraction T. Then f (T ) sup z <1 f (z). it also works on the bidisk [Ando, 1963]. It does not always work beyond these cases [Varapoulos 1974], [Crabb & Davie 1975], [Arveson, 1998] The Schur class S consists of analytic functions f on B with f := sup f (z) 1. (2) z B The Schur Agler class SA S consists of functions s.t. f A := sup f (T ) 1, (3) T taken over commuting tuples T = (T (r) ) r=1,...,k;i=1,...,lr,j=1,...,m r with (T (r) lr mr ) i=1,j=1 < 1. Introduced on polydisk by [Agler, 1990]. Later generalizations [Ambrozie & Timotin, 2003], [Ball & Bolotnikov, 2004].

4 A polydisk result If p(z) is a stable (i.e., no roots on D d ) polynomial, then z n p(1/z) p(z) S d whenever n degp. (4) For instance, z 1z 2 z 3 (z 2 z 3 + z 1 z 3 + z 1 z 2 )/3. 1 (z 1 + z 2 + z 3 )/3 By [Rudin, 1969] every rational inner function is of the form (4). Theorem Let p admit representation (1) for some n and contractive K. Then z n p(1/z) p(z) = det[(z n K )(I KZ n ) 1 ], Z n = d j=1 z ji nj (5) belongs to the Schur-Agler class SA d. [G,K-V,W 2013] Generalized to square-matrix polyball in [G,K-V,V,W 2015]

5 A non Schur-Agler rational inner function Consider the Kaser Varopoulos polynomial p(z 1, z 2, z 3 ) = 1 5( z z z2 3 2z 1z 2 2z 2 z 3 2z 3 z 1 ), which satisfies p = 1, deg p = deg p = (2, 2, 2), and there exist commuting contractions T 1, T 2, T 3 [Holbrook, 2001] s. t. p(t 1, T 2, T 3 ) = 6/5 and T 1 T 2 T 3 = 0. For 0 < r 1, q(z) = 1 + rz (3,3,3) p(1/z) = 1 + rz 1 z 2 z 3 p (z) is stable, and the function f (z) = q (z) q(z) = z3 1 z3 2 z3 3 + rp(z) 1 + rz 1 z 2 z 3 p (z) belongs to S 3. Since f (T ) = rp(t ), f is not in SA 3 whenever 5 6 < r 1. Hence, q(z) det(i KZ (3,3,3)), with K 1. [G,K-V,W 2013]

6 Main result Theorem [G, K-V, V, W, 2015] Let p be a polynomial in the commuting indeterminates z (r), r = 1,..., k, i = 1,..., l r, j = 1,..., m r, with p(0) = 1, which is strongly stable (=nonzero on closure) with respect to the matrix polyball B l 1 m 1 B l k m k. Then there exist n = (n 1,..., n k ) Z k + and a strict contraction K C k r=1 mr nr k r=1 lr nr so that p(z) = det(i KZ n ), (6) where Z = k r=1 (Z r I nr ) and Z r = (z (r) ) C lr mr. For bidisk stability suffices and n = degp [G, K-V, V, W, 2014]. Corollary For every rational inner f S d with strongly stable denominator, there exists an n so that z n f (z) SA d. [G, K-V, V, W, 2015]

7 Main ideas in proof Since (irreducible) p has no zeros in B, for some ρ > 1, g ρ (z) = 1 p(ρz) satisfies g ρ A <. Let 1 c > g ρ A. By matrix valued Positivstellensatz [G,K-V,V,W 2015] find a n = (n 1,..., n k ) Z k + and a contractive matrix [ ] A B C D s. t. k cg ρ = D + CZ (I AZ ) 1 B, Z = (Z r I nr ). r=1 Lift the rational function cg ρ to a noncommutative rational expression using the same realization formula, D + Cζ(I Aζ)) 1 B, now with ζ = (ζ (r) ) being noncommuting indeterminates Use [Ball, Groenewald, Malakorn 2005] to compress to minimal realization. Use minimality to show that singularities of (cg ρ ) 1 coincide with det(i A min Z ) = 0, giving det(i A min Z ) 1.

8 Square-matrix polyball (thus l r = m r, r = 1,..., k). Theorem A scalar-valued function f on B is rational inner (i.e., f = 1 on regular points of the distinguished boundary (=unitary Z )) if and only if there exist a B-stable polynomial p and nonnegative integers m 1,..., m k such that f (Z ) = k p (Z ) (det Z (r) ) mr p(z ). (7) r=1 One can choose p to be coprime with p. [Korányi, Vagi 1977] Corollary Let f be a rational inner function on B which is regular on B. Then there exist nonnegative integers s 1,..., s k such that k r=1 (det Z (r) ) sr f SA. [G, K-V, V, W, 2015]

9 Our papers A. Grinshpan, D. S. Kaliuzhnyi-Verbovetskyi, and H. J. Woerdeman. Norm-constrained determinantal representations of multivariable polynomials. Complex Anal. Oper. Theory 7 (2013), The Schwarz lemma and the Schur-Agler class. Proceedings of MTNS A. Grinshpan, D. S. Kaliuzhnyi-Verbovetskyi, V. Vinnikov, and H. J. Woerdeman. Stable and real zero polynomials in two variables, Multidim. Syst. Signal Proc., 2014, Matrix-valued Hermitian Positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials, ArXiv: Contractive determinantal representations of stable polynomials on a matrix polyball, ArXiv: Rational inner functions on a square-matrix polyball, ArXiv: THANK YOU!