Rational inner functions on a square-matrix polyball

RATIONAL INNER FUNCTIONS ON A SQUARE-MATRIX POLYBALL

arXiv:1505.05437v1 [math.CV] 20 May 2015

ANATOLII GRINSHPAN, DMITRY S. KALIUZHNYI-VERBOVETSKYI, VICTOR VINNIKOV, AND HUGO J. WOERDEMAN Dedicated to the blessed memory of Cora Sadosky, our dear friend and colleague. Abstract. We establish the existence of a finite-dimensional unitary realization for every matrix-valued rational inner function from the Schur–Agler class on a unit square-matrix polyball. In the scalar-valued case, we characterize the denominators of these functions. We also show that every polynomial with no zeros in the closed domain is such a denominator. One of our tools is the Kor´ anyi–Vagi theorem generalizing Rudin’s description of rational inner functions to the case of bounded symmetric domains; we provide a short elementary proof of this theorem suitable in our setting.

1. Introduction In this paper, we study rational inner functions on the Cartesian product of square-matrix Cartan’s domains of type I, i.e., on a unit square-matrix polyball, B = Bℓ1 ×ℓ1 × · · · × Bℓk ×ℓk n o = Z = (Z (1) , . . . , Z (k) ) ∈ Cℓ1 ×ℓ1 × · · · × Cℓk ×ℓk : kZ (r) k < 1, r = 1, . . . , k . Lk (r) with kZk < 1. We can interpret points of B as block-diagonal matrices Z = r=1 Z Then B is a special case of domain DP defined by a matrix polynomial P as a set of z = (z1 , . . . , zd ) ∈ Cd satisfying kP(z)k < 1 (see [2, 4, 11]); in this case, P = Z viewed as Pk 2 (r) a polynomial in matrix entries zij , i, j = 1, . . . , ℓr , r = 1, . . . , k, and d = r=1 ℓr . In particular, B is a special case of a Cartesian product of (not necessarily square) matrix Cartan’s domains of type I (see [14, 10]). The distinguished (or Shilov) boundary of B consists of k-tuples of unitary matrices, n o ∂S B = Z = (Z (1) , . . . , Z (k) ) ∈ Cℓ1 ×ℓ1 × · · · × Cℓk ×ℓk : Z (r)∗ Z (r) = I, r = 1, . . . , k ,

which can also be interpreted as a set of block-diagonal unitary matrices. Notice that the unit polydisk Dd is a special case of a unit square-matrix polyball where k = d, and ℓr = 1 for (r) r = 1, . . . , k. We consider matrix functions in variables zij . We denote the corresponding dtuple of variables by z and, for a function F , we identify F (z) = F (Z). An s×s matrix-valued (r) function F is rational inner if each matrix entry Fαβ is a rational function in d variables zij which is regular in B, and F takes unitary matrix values at each of its regular points on 2010 Mathematics Subject Classification. 26C10, 32A10, 47A48, 47N70, 93B15. Key words and phrases. Rational inner function; matrix polyball; classical Cartan domains; unitary realization; stable polynomial; contractive determinantal representation; eventual Agler denominators. AG, DK-V, HW were partially supported by NSF grant DMS-0901628. DK-V and VV were partially supported by BSF grant 2010432. 1

the distinguished boundary ∂S B. Notice that the zero variety of the least common multiple of the denominators of the rational functions Fαβ in their coprime fraction representation has an intersection with ∂S B of relative Lebesgue measure zero, which can be proved using an argument analogous to that of [6, Lemma 6.3]; thus almost all points of ∂S B are regular points of F . Define n

TZ = T = T

k M

T (r) :

r=1

(r)

=

(r) r , [Tij ]ℓij=1

(r) Tij

o are commuting operators on a Hilbert space and kT k < 1 .

For T ∈ TZ , the Taylor joint spectrum [19] of T viewed as a multioperator (r)

(Tij )r=1,...,k; i,j=1,...,ℓr lies in the domain B, and for a matrix-valued function F analytic on B one can define F (T ) by means of Taylor’s functional calculus [20]; see [2] and a further discussion in [4]. We say that an s × s matrix-valued function F analytic on B belongs to the Schur–Agler class SAZ (Cs ) associated with B, or rather with its defining polynomial P = Z, if its associated Agler norm, kF kA,Z = sup kF (T )k T ∈TZ

is at most 1. In the scalar case, s = 1, we simply write SAZ . In the case of the unit polydisk Dd , this class coincides with the classical Schur–Agler class SAd studied in the seminal paper of Agler [1]. Notice that SAZ (Cs ) is a subclass of the Schur class SZ (Cs ) of s×s matrix-valued contractive analytic functions on B. It follows from [21] and [3] that these classes do not coincide, i.e., the analog of von Neumann’s inequality fails when d ≥ 2, unless B = D2 . Moreover, rational inner functions, which obviously belong to the Schur class, do not necessarily belong to the Schur–Agler class: an example of a rational inner function on D3 which is not Schur–Agler was given in [12, Example 5.1]. In Section 2, we give a characterization of scalar-valued rational inner functions on B in terms of their coprime fraction representation. In Section 3, we describe matrix-valued rational inner functions on B that belong to the associated Schur–Agler class as the functions that have a finite-dimensional unitary realization. In Section 4, we characterize eventual Agler denominators, i.e., those multivariable polynomials which can be represented as the denominators of scalar rational inner functions in the Schur–Agler class SAZ , in terms of certain contractive determinantal representations. We also show that every polynomial with no zeros on the closed domain, B, is an eventual Agler denominator, and we end with several open questions. 2. Scalar-valued rational inner functions (r)

For a polynomial p in d variables zij , i, j = 1, . . . , ℓr , r = 1, . . . , k, where d = define its reverse with respect to B as ← − p (Z) =

k Y

(det Z (r) )tr p(Z ∗−1 ),

r=1

2

Pk

2 r=1 ℓr ,

we

(r)

where tr is the total degree of p in the variables zij , i, j = 1, . . . , ℓr . We say that a polynomial p is B-stable (resp., strongly B-stable) if p does not have zeros in B (resp., in B). The following result is a generalization of Rudin’s characterization of rational inner functions on the polydisk [18] to the case of a square-matrix polyball B. It appears in more generality in [17, Theorem 3.3], where Rudin’s theorem is extended to all bounded symmetric domains. We provide a proof that applies to the specific setting of B, and therefore requires less machinery. Theorem 2.1. A scalar-valued function f on B is rational inner if and only if there exist a B-stable polynomial p and nonnegative integers m1 , . . . , mk such that k ← − Y p (Z) f (Z) = (det Z (r) )mr . p(Z) r=1

(2.1)

− One can choose p to be coprime with ← p. For the proof of Theorem 2.1, we will need the following proposition. Proposition 2.2. Let p be a B-stable polynomial, and suppose that |p(Z)| = 1 for all Z ∈ ∂S B. Then there exist nonnegative integers m1 , . . . , mk such that k Y p(Z) = (det Z (r) )mr . r=1

Proof. Notice that if Z (r) is unitary for each r, then ← − p (Z)p(Z) =

(2.2)

k Y (det Z (r) )tr . r=1

Since ∂S B is a uniqueness set for analytic functions (see, e.g., [14]), we have that (2.2) holds for all Z = (Z (1) , . . . , Z (k) ) ∈ Cℓ1 ×ℓ1 ×· · ·×Cℓk ×ℓk . Since det Z (r) is an irreducible polynomial Q (r) in matrix entries zij (see, e.g., [7, Section 61]) we obtain that p(Z) = kr=1 (det Z (r) )mr for some mr ≤ tr , r = 1, . . . , k.  Proof of Theorem 2.1. The sufficiency of the representation (2.1) for f to be rational inner is clear. To prove the necessity, we first write f = q/p, with p and q coprime. Since f is analytic in B, we have that p is B-stable. Next, for Z ∈ ∂S B we have that q(Z)q(Z −1∗ ) = p(Z)p(Z −1∗ ). Hence the equality k Y

(r) τr

(det Z )

q(Z)q(Z −1∗ )

=

k Y

(det Z (r) )τr p(Z)p(Z −1∗ )

r=1

r=1

(r)

holds for every Z ∈ ∂S B, where τr is the maximum of the total degrees in zij , i, j = 1, . . . , ℓr , in q and p. Then it also holds for all Z. Since q and p are coprime, p is a divisor of Q k (r) τr ) q(Z −1∗ ). Hence r=1 (det Z k Y (det Z (r) )τr p(Z −1∗ ), u(Z)q(Z) = r=1

3

for some polynomial u. Observe that |u(Z)| = 1 on ∂S B. Then by Proposition 2.2 we obtain Q that u(Z) = kr=1 (det Z (r) )µr , with some µr ≤ τr , and k Y (det Z (r) )τr −µr p(Z −1∗ ). q(Z) = r=1

The latter equality implies that mr := τr − µr − tr ≥ 0, where tr is the total degree of p in (r) variables zij , i, j = 1, . . . , ℓr . It follows that k Y − q(Z) = (det Z (r) )mr ← p (Z), r=1

and (2.1) holds.



Remark 2.3. Proposition 2.2 and Theorem 2.1 also hold when B is replaced by a Cartesian product of square-matrix Cartan’s domains of type II [14], n o Bsym = Z = (Z (1) , . . . , Z (k) ) ∈ B : Z (r) = Z (r)⊤ , r = 1, . . . , k ,

or by a Cartesian product of mixed type involving Cartan’s domains of types I and II. The (r) proofs are similar after noticing that det Z (r) as a polynomial in zij , i, j = 1, . . . , ℓr : i ≤ j, is also irreducible when Z (r) = Z (r)⊤ ; see, e.g., [7, Section 61]. 3. Matrix-valued rational inner functions from the Schur–Agler class We now characterize matrix-valued rational inner functions on the unit square-matrix polyball B, which belong to the associated Schur–Agler class, in terms of their unitary realizations. This is a generalization of the result [5, Theorem 2.1] for the unit polydisk Dd which, in turn, is a matrix-valued extension of the result from [16] in the scalar-valued setting (see also an earlier paper [9] for the bidisk case). Theorem 3.1. An s × s matrix-valued function F on B is rational inner and belongs to the class SAZ (Cs ) if and only if F has a finite-dimensional unitary realization, i.e., there exist nonnegative integers n1 , . . . , nk and a unitary matrix   Pk Pk A B ∈ C( r=1 ℓr nr +s)×( r=1 ℓr nr +s) C D such that F (Z) = D + CZn (I − AZn )−1 B, L where Zn = kr=1 (Z (r) ⊗Inr ). If we write F = QP −1 , where P and Q are matrix polynomials of total degree at most g such that P ∗ P = Q∗ Q on ∂S B, then n1 , . . . , nk can be chosen so
that nr ≤ lr s g+d−1 , r = 1, . . . , k. d For the proof of Theorem 3.1, we will need the following proposition. Recall that a linear mapping Φ : Ca×a → Cb×b is said to be completely positive if, for every m ∈ N, the mapping Φ(m) : (Ca×a )m×m → (Cb×b )m×m defined by (Φ(m) (A))ij = Φ(Aij ), i, j = 1, . . . , m, is positive, that is, it maps every positive semidefinite matrix A to a positive semidefinite matrix Φ(m) (A). 4

Proposition 3.2 ([8, Theorem 1]). Let Φ : Ca×a → Cb×b be a completely positive linear 2 mapping. Then there exists Y ∈ Ca b×b so that Φ(X) = Y ∗ (X ⊗ Iab )Y . Proof of Theorem 3.1. The sufficiency part is analogous to that of [6, Theorem 6.1]. To prove the necessity, let F = QP −1 , where P and Q are matrix polynomials of total degree at most g, P ∗ P = Q∗ Q on ∂S B, and assume that F ∈ SAZ (Cs ). Then by [4, Theorem 1.5] there exist separable Hilbert spaces Kr and analytic functions Hr on B with values linear operators from Cs to Cℓr ⊗ Kr such that k   X ∗ (r)∗ (r) ∗ ∗ P (W ) P (Z) − Q(W ) Q(Z) = Hr (W ) (I − W Z ) ⊗ IKr Hr (Z), Z, W ∈ B. r=1

Letting Z = W = tU where |t| < 1 and U ∈ ∂S B, we obtain k

(3.1)

P (tU)∗ P (tU) − Q(tU)∗ Q(tU) X ∗ = Hr (tU)Hr (tU). 1 − |t|2 r=1

Since P (tU)∗ P (tU) = Q(tU)∗ Q(tU) for all U = (U (1) , . . . , U (r) ) ∈ ∂S B and |t| = 1, the numerator of the left-hand side of (3.1) is a polynomial in t and t which vanishes on the variety 1 − tt = 0. Therefore the left-hand side of (3.1) is a polynomial in t and t and a (r) trigonometric polynomial in matrix entries uij , i, j = 1, . . . , ℓr , r = 1, . . . , k. Let Pα and Qα be the coefficients of z α in the polynomials P and Q, respectively, and let Hr,α be the coefficient of z α in the Maclaurin series for Hr , where for z = (z1 , . . . , zd ) and α = (α1 , . . . , αd ) we set z α = z1α1 · · · zdαd . Then the zeroth Fourier coefficient of the left-hand side of (3.1) as (r) a trigonometric polynomial in variables uij is X 1 (Pα∗ Pα − Q∗α Qα )|t|2|α| , 2 1 − |t| |α|≤g

where |α| = α1 +· · ·+αd . Note that the preceding expression is a polynomial in |t|2 of degree at most g − 1. The zeroth Fourier coefficient of the right-hand side of (3.1) (for sufficiently small t), k X X ∗ Hr,α Hr,α|t|2|α| , r=1

α

2

∗ is therefore a polynomial in |t| as well, and Hr,α Hr,α = 0 for |α| ≥ g. g+d−1 g+d−1 Consider now the completely positive map Φr : Cℓr ×ℓr → Cs( d )×s( d ) defined via ∗ Φ(X) = col|α|≤g−1 (Hr,α )(X ⊗ IKr )row|α|≤g−1 (Hr,α ). 2

a b×b Then by Proposition 3.2 we can such that Φr (X) = Yr∗ (X ⊗ Iab )Yr ,
find matrices Yr ∈ C g+d−1 where a = ℓr and b = s d . Writing Yr = row|α|≤g−1 (Yr,α ), we can form a polynomial X Gr (Z) = Yr,αz α |α|≤g−1


with the coefficients in Cℓr nr ×s , where nr = ℓr s g+d−1 , so that d     (r)∗ (r) ∗ (r)∗ (r) ∗ Hr (W ) (I −W Z )⊗IKr Hr (Z) = Gr (W ) (I −W Z )⊗Inr Gr (Z), 5

r = 1, . . . , k,

and (3.2)

∗

∗

P (W ) P (Z) − Q(W ) Q(Z) =

k X r=1

  Gr (W )∗ (I − W (r)∗ Z (r) ) ⊗ Inr Gr (Z).

Rearranging the terms in (3.2), we obtain k k   X X ∗ ∗ ∗ (r)∗ (r) Gr (W )∗ Gr (Z). P (W ) P (Z) + Gr (W ) W Z ⊗ Inr Gr (Z) = Q(W ) Q(Z) + r=1

r=1

Therefore

   (Z (1) ⊗ In1 )G1 (Z) G1 (Z) ..    .  .   h 7→  ..  h (Z (k) ⊗ I )G (Z) G (Z) 

nk

k

k

Q(Z) P (Z) is a linear and isometric map from the span of the elements on the left to the span  of the A B elements on the right. It may be extended (if necessary) to a unitary matrix so that C D AZn G(Z) + BP (Z) = G(Z) CZn G(Z) + DP (Z) = Q(Z), Lk

for every Z; here Zn = r=1 (Z (r) ⊗ Inr ). Solving the first equation above for G(Z) and then plugging the result into the second equation yield F (Z) = Q(Z)P −1 (Z) = D + CZn (I − AZn )−1 B.  Remark 3.3. An analog of Theorem 3.1, with a similar proof, is also valid for Cartesian products of Cartan’s domains of type II or III [14] or for Cartesian products of mixed type involving Cartan’s domains of types I, II, and III. We recall here that a Cartan domain of type II (resp., III) is a (lower-dimensional) subset of a square-matrix Cartan’s domain of type I consisting of symmetric (resp., antisymmetric) matrices. 4. Eventual Agler denominators (r)

We will say that a polynomial v in zij , i, j = 1, . . . , ℓr , r = 1, . . . , k, is almost self-reversive − with respect to the square-matrix polyball B if ← v = γv, for some scalar γ with |γ| = 1. We have the following generalization of a result that was announced in [13] for the case of a unit polydisk. − Theorem 4.1. Let p be a B-stable polynomial which is coprime with ← p . Then the following are equivalent: (i) p is an eventual Agler denominator, that is, there exist nonnegative integers s1 , . . . , sk Q − such that the rational inner function kr=1 (det Z (r) )sr ← p (Z)/p(Z) is in the Schur– Agler class SAZ . (ii) There exists an almost self-reversive polynomial v of multidegree (s1 , . . . , sk ) such that p(Z)v(Z) = det(I − KZn ) for some nonnegative integers nr , r = 1, . . . , k, and L a contractive matrix K, where Zn = kr=1 (Z (r) ⊗ Inr ). 6

Q − Proof. (i)⇒(ii) Let f (Z) = kr=1 (det Z (r) )sr ← p (Z)/p(Z) be in SAZ . By Theorem 3.1 there B ] such exists a k-tuple n = (n1 , . . . , nk ) of nonnegative integers and a unitary matrix [ CA D that f (Z) = D + CZn (I − AZn )−1 B. Observe that

  I − AZn B det −CZn D −1 f (Z) = D + CZn (I − AZn ) B = det(I − AZn )

and     ∗    I − AZn B A B A 0 Zn 0 det = det det − = λ det(A∗ − Zn ), −CZn D C D B∗ 0 0 −1 

B ]. Hence where λ = det [ CA D

(4.1)

k Y − (det Z (r) )sr ← p (Z) det(I − AZn ) = λp(Z) det(A∗ − Zn ). r=1

Q − − p do not Since p is B-stable and coprime with ← p , the polynomials p and kr=1 (det Z (r) )sr ← have common factors. Therefore p divides det(I − AZn ), i.e., there exists a polynomial v so that p(Z)v(Z) = det(I − AZn ). Then det Zn p(Z ∗−1 )v(Z ∗−1 ) = det(Zn − A∗ ). Using (4.1), we obtain k Y r=1

− p (Z)p(Z)v(Z) = λ (det Z (r) )sr ←

k Y

Pk − (det Z (r) )nr −degr p p(Z)← p (Z)v(Z ∗−1 )(−1) r=1 ℓr nr ,

r=1

(r)

where degr p is the total degree of p in the variables zij . After dividing out we see that v is almost self-reversive and sr = nr − degr p − degr v. Clearly, K = A is a contractive matrix. (ii)⇒(i) Suppose there exists an almost self-reversive polynomial v such that p(Z)v(Z) = det(I − KZn ) with a contractive matrix K and a k-tuple n = (n1 , . . . , nk ) of nonnegative integers. Then by a straightforward modification of [12, Theorem 5.2] the rational inner function Qk (r) nr ∗−1 )v(Z ∗−1 ) r=1 (det Z ) p(Z p(Z)v(Z) is Schur–Agler. Since v is almost self-reversive, the rational inner function Qk (r) nr −degr v p(Z ∗−1 ) ) r=1 (det Z p(Z) is Schur–Agler, i.e., p is an eventual Agler denominator.



Remark 4.2. An analog of Theorem 4.1 is valid for a Cartesian product of Cartan’s domains of type II, i.e., for a domain Bsym , or for a Cartesian product of mixed type involving Cartan’s domains of types I and II; see Remark 2.3 and Remark 3.3. 7

The following result is a generalization of [10, Corollary 3.2] formulated for the polydisk Dd to the case of the square-matrix polyball B. Theorem 4.3. Every strongly B-stable polynomial is an eventual Agler denominator. Proof. By [10, Theorem 3.1] there exist Pa k-tuple P of nonnegative integers n = (n1 , . . . , nk ) ( kr=1 ℓr nr )×( kr=1 ℓr nr ) and a strictly contractive matrix K ∈ C such that p(Z) = det(I − KZn ), Lk (r) where Zn = r=1 (Z ⊗ Inr ). Then, similarly to the last paragraph in the proof of Theorem 4.1 (with v = 1), one shows that p is an eventual Agler denominator.  Corollary 4.4. Let f be a rational inner function is regular on ∂S B. Then there Qk on B which (r) sr exist nonnegative integers s1 , . . . , sk such that r=1 (det Z ) f ∈ SAZ . − Proof. By Theorem 2.1 there exists a stable polynomial p which is coprime with ← p and such that (2.1) holds with some nonnegative integers m1 , . . . , mr . Since f is regular on ∂S B, the polynomialQp is strongly B-stable. By Theorem 4.3, p is an eventual Agler denominator.  Therefore kr=1 (det Z (r) )sr f ∈ SAZ for some nonnegative integers s1 , . . . , sk .

The question on whether the assumption of regularity of f on ∂S B in Corollary 4.4 can be removed is open. Another open question is whether Corollary 4.4 holds for matrixvalued rational inner functions. Finally, it is interesting to investigate the analogues of the results in this paper for the unbounded version of the domain B, i.e., the Cartesian product of matrix halfplanes. The Cayley transform over the matrix variables Z (r) , r = 1, . . . , k, would allow one to obtain a finite-dimensional realization formula for rational inner functions on the product of matrix halfplanes; if, in addition, the Cayley transform over the values of a function is applied, then one can obtain the corresponding realization formula for rational Cayley inner functions over the product of matrix halfplanes (see [5] for the case of a polyhalfplane, i.e., the product of scalar halfplanes). We would also like to mention [15] where a subclass of Cayley inner functions on the product of matrix halfplanes, the Bessmertnyi class, was studied. References

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Department of Mathematics, Drexel University, 3141 Chestnut St., Philadelphia, PA, 19104 E-mail address: {tolya,dmitryk,hugo}@math.drexel.edu Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel, 84105 E-mail address: [email protected]

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