UNITARY INTERPOLANTS AND FACTORIZATION INDICES OF MATRIX FUNCTIONS 1 R.B. ALEXEEV AND V.V. PELLER 0 0 2 n a Abstract. For an n×n bounded matrix function Φ we study unitary inter- J polants U, i.e., unitary-valued functions U such that Uˆ(j) = Φˆ(j), j < 0. We 6 are looking for unitary interpolants U for which the Toeplitz operator T is 2 U Fredholm. We give a new approach based on superoptimal singular values and ] thematic factorizations. We describe Wiener–Hopf factorization indices of U in A termsofsuperoptimalsingularvaluesofΦandthematic indicesofΦ−F,where F F is a superoptimal approximation of Φ by bounded analytic matrix functions. . h The approachessentially relies on the notion of a monotone thematic factoriza- t tion introduced in [AP]. In the last section we discuss hereditary properties of a m unitary interpolants. In particular, for matrix functions Φ of class H∞+C we study unitary interpolants U of class QC. [ 1 v 1. Introduction 4 2 2 1 A Hankel operator defined on the Hardy class H2(Cn) of Cn-valued functions 0 has infinitely many different symbols. If Φ is a symbol of such a Hankel operator, 1 0 then Φ − Q is a symbol of the same Hankel operator for any bounded analytic / matrix function Q. A natural problem arises for a Hankel operator to choose a h t symbol that satisfies certain nice properties. For example, an important problem is a m to choose a symbol that has minimal L∞-norm. In this paper we consider another : important problem to choose a symbol that takes unitary values (such symbols are v called unitary-valued). i X Recall that for a bounded M -valued function Φ (we denote by M the space m,n m,n ar ofm×nmatrices)ontheunitcircleTtheHankel operatorHΦ : H2(Cn) → H−2(Cm) with symbol Φ is defined by H f d=ef P Φf, f ∈ H2(Cn), Φ − where P is the orthogonal projection onto H2(Cm) d=ef L2(Cm)⊖H2(Cm). − − Certainly, when we discuss the problem of finding unitary-valued symbols we have to assume that m = n, i.e., Φ ∈ L∞(M ). If U is a symbol of H , then the n,n Φ The second author is partially supported by NSF grant DMS 9970561. 1 Fourier coefficients of U satisfy Uˆ(j) = Φˆ(j), j < 0. A unitary-valued matrix function U satisfying this condition is called a unitary interpolant of Φ. A matrix analog of Nehari’s theorem says that kHΦk = distL∞ Φ,H∞(Mm,n) (1.1) (see [Pa]). Here for a bounded Mm,n-valu(cid:0)ed function F(cid:1)we use the notation def kFkL∞ = esssupkF(ζ)kMm,n, ζ∈T where the norm of a matrix in M is its operator norm from Cn to Cm. m,n Recall that by Hartman’s theorem, H is compact if and only if Φ Φ ∈ (H∞ +C)(M ) (see e.g., [N]), where m,n H∞ +C d=ef {f +g : f ∈ H∞, g ∈ C(T)}, while the essential norm kH k (i.e., the distance from H to the set of compact Φ e Φ operators) can be computed as follows: kHΦke = distL∞ Φ,(H∞ +C)(Mm,n) (see e.g., [Sa] for the proof of this formu(cid:0)la for scalar function(cid:1)s, in the matrix-valued case the proof is the same). It follows from (1.1) that for a matrix function Φ ∈ L∞(M ) to have a unitary n,n interpolant it is necessary that kH k ≤ 1. Φ If ϕ is a scalar function such that kH k ≤ 1, then a unitary interpolant of ϕ ϕ exists if H has two different symbols whose L∞-norms are at most 1 (the ϕ Adamyan–Arov–Krein theorem [AAK2]). Dym and Gohberg studied in [DG1] the problem of finding unitary interpolants for matrix functions with entries in a Banach algebra X of continuous functions on T that satisfy certain axioms (their axioms are similar to axioms (A1)–(A4) in §5). They showed that if Φ belongs to the space X(M ) of M -valued functions n,n n,n with entries in X and kH k ≤ 1, then Φ has a unitary interpolant U of the same Φ class X(M ). Moreover, the negative indices of a Wiener–Hopf factorization of n,n U are uniquely determined by Φ while the nonnegative indices can be arbitrary. Note, however, that we were not able to understand their argument in the proof of Lemma 4.10. In [DG2] the problem of finding unitary interpolants was studied in a more general situation. Another approach to this problem was given by Ball in [B]. In this paper we propose a different approach based on so-called thematic factorizations introduced in [PY1]. We state our results in different terms. We obtain information about the Wiener–Hopf indices of unitary interpolants of Φ in terms of 2 the so-called superoptimal singular values of Φ and indices of thematic factorizations of Φ−F, where F is a superoptimal approximation of Φ by bounded analytic matrix functions (see §2). We are interested in unitary interpolants U of Φ such that the Toeplitz operator T is Fredholm (i.e., invertible modulo the compact operators). U Recall that for Ψ ∈ L∞(M ) the Toeplitz operator T : H2(Cn) → H2(Cm) is m,n Ψ defined by T f = P Ψf, f ∈ H2(Cn), Ψ + where P is the orthogonal projection onto H2(Cm). + By Simonenko’s theorem [Si] (see also [LS]), the symbol Ψ ∈ L∞(M ) of a n,n Fredholm Toeplitz operator T admits a Wiener–Hopf factorization: Ψ zd0 0 ··· 0 0 zd1 ··· 0 Ψ = Q∗2 ... ... ... ... Q1,  0 0 ··· zdn−1      where d ,··· ,d ∈ Z are Wiener–Hopf indices, and Q and Q are functions 0 n−1 1 2 invertible in H2(M ). Here we start enumeration with 0 for technical reasons. n,n Clearly, we can always arrange the indices in the nondecreasing order: d ≤ d ≤ ··· ≤ d 0 1 n−1 inwhich casetheindices d areuniquely determinedbythefunctionΨ. Uniqueness j follows easily from the following well-known identity dimKerT = −d (1.2) Ψ j {j:Xdj<0} applied to the matrix functions zjΨ, j ∈ Z. Let U be a unitary-valued function on T. It is well known that the Toeplitz operator T is Fredholm if and only if U kHUke < 1, and kHU∗ke < 1. (1.3) Indeed, suppose that T is Fredholm. Clearly, kH k = kH k for any j ∈ Z. U zjU e U e Multiplying U by zN if necessary, we may assume that KerT is trivial, and so T U U is left-invertible. It follows now from the obvious equality kH fk2 +kT fk2 = kfk2, f ∈ H2(Cn), U 2 U 2 2 that kH k < 1, and so kH k < 1. Applying the same reasoning to U∗, we find U U e that kHU∗ke < 1. Suppose now that (1.3) holds. It is easy to see that TU∗TU = I −HU∗HU, TUTU∗ = I −HU∗∗HU∗, 3 Since kHUke < 1 and kHU∗ke < 1, it follows that TU∗TU and TUTU∗ are invertible modulo the compact operators which implies that T is Fredholm. U Obviously, if U is a unitary interpolant of Φ, then H = H . Therefore for Φ U Φ to have a unitary interpolant U with Fredholm T it is necessary that kH k < 1. U Φ e Throughout this paper we assume that this condition is satisfied. We conclude the introduction with the following fact. Lemma 1.1. Let U be a unitary-valued matrix function in L∞(M ) such that n,n kH k < 1. Suppose that U admits a Wiener–Hopf factorization of the form U e U = Q∗ΛQ , 2 1 where Q and Q are invertible in H2(M ) and Λ is a diagonal matrix function 1 2 n,n with diagonal terms zdj. Then T is Fredholm and U kHUke = kHU∗ke. Proof. Multiplying U by z¯N for a sufficiently large N if necessary, we may assume that all exponents d are nonpositive. Let us show that in this case T has j U dense range. Suppose that g ∈ H2(Cn) and g ⊥ RangeT . Let f be a polynomial U in H2(Cn). We have 0 = (T f,g) = (Uf,g) = (Q f,Λ∗Q g). U 1 2 Clearly, Λ∗ ∈ H∞(M ) and the set of functions of the form Q f is dense in n,n 1 H2(Cn). Hence, Λ∗Q g = 0, and so g = 0 which proves that T has dense range. 2 U In [Pe3] it was proved that if T has dense range in H2(Cn), then the operator U HU∗∗HU∗ is unitarily equivalent to the restriction of the operator HU∗HU to the subspace H2(Cn)⊖{f ∈ H2(Cn) : kH fk = kfk } U 2 2 (see also [PK] where this was proved in the scalar case). Clearly, the condition kH k < 1 implies that the above subspace has finite codimension. Therefore U e kHU∗ke = kHUke < 1 and as we have already observed, this implies that TU is (cid:4) Fredholm. To simplify the notation, for a matrix function Φ and a space X of functions on T we can write Φ ∈ X when all entries of Φ belong to X, if this does not lead to confusion. In §2 we collect necessary information on superoptimal approximation and thematic (as well as partial thematic) factorizations. In particular, we define the important notion of a monotone (partial) thematic factorization that was introduced in [AP] and state the theorem on the invariance of indices of monotone (partial) thematic factorizations obtained in [AP]. In §3 we study unitary interpolants of matrix functions Φ ∈ L∞(M ) satisfying n,n kH k < 1. We state the main results in terms of the superoptimal singular values Φ e 4 of Φ and the indices of a monotone (partial) thematic factorization of Φ−F, where F is a best approximation of Φ by bounded analytic functions. Finally,weshowin§4howtoapplytheresultsof§3tostudyunitaryinterpolants of class X for various function spaces X. In particular we show in §4 that if Φ ∈ (H∞+C)(M ), then all unitary interpolants U of Φ satisfying the condition n,n kHU∗ke < 1 belong to the class QC d=ef {f ∈ H∞ +C : f¯∈ H∞ +C}. 2. Superoptimal approximation and thematic indices This section is an introduction to superoptimal approximation and thematic factorizations. We refer the reader to [PY1], [PY2], [PT], and [AP] for more detailed information. It is a well-known fact [Kh] that if ϕ is a scalar function of class H∞ +C, then there exists a unique function f ∈ H∞ such that distL∞(ϕ,H∞) = kϕ−fkL∞. ForamatrixfunctionΦ ∈ L∞(M )wesaythatamatrixfunctionF ∈ H∞(M ) m,n m,n is a best approximation of Φ by bounded analytic matrix functions if kΦ−FkL∞ = distL∞(Φ,H∞(Mm,n). However, consideration of diagonal matrices makes it obvious that for matrix functions we can have uniqueness only in exceptional cases. To define a superoptimal approximation of a matrix function Φ ∈ L∞(M ) by m,n bounded analytic matrix functions, we consider the following sets: Ω = {F ∈ H∞(M ) : F minimizes t d=ef esssupkΦ(ζ)−F(ζ)k}; 0 m,n 0 ζ∈T def Ω = {F ∈ Ω : F minimizes t = esssups (Φ(ζ)−F(ζ))}. j j−1 j j ζ∈T Recall that for a matrix A ∈ M the jth singular value s (A) is defined by m,n j s (A) = inf{kA−Rk : rankR ≤ j}, j ≥ 0. j Functions in F ∈ Ω are called superoptimal approximations of Φ by min{m,n}−1 analytic functions, or superoptimal solutions of Nehari’s problem. The numbers t j are called superoptimal singular values of Φ. In [PY1] it was shown that any matrix function Φ ∈ (H∞ + C)(M ) has a m,n unique superoptimal approximation by bounded analytic matrix functions. An- other approach to the uniqueness of a superoptimal approximation was found later in [T]. Those results were extended in [PT] to the case of matrix functions 5 Φ ∈ L∞(M ) such that the essential norm kH k of the Hankel operator H is m,n Φ e Φ less than the smallest nonzero superoptimal singular value of Φ. A matrix function Φ ∈ L∞(M ) is called badly approximable if m,n distL∞(Φ,H∞(Mm,n)) = kΦkL∞. It is called very badly approximable if the zero matrix function is a superoptimal approximation of Φ. Recall that a nonzero scalar function ϕ ∈ H∞ + C is badly approximable if and only if it has constant modulus almost everywhere on T, belongs to QC, and indT > 0. For continuous ϕ this was proved in [Po] (see also [AAK1]). For the ϕ general case see [PK]. More generally, if ϕ is a scalar function in L∞ such that kH k < kH k, then ϕ is badly approximable if |ϕ| is constant almost everywhere ϕ e ϕ on T, T is Fredholm, and indT > 0. ϕ ϕ Let us now define a thematic matrix function. Recall that a function F ∈ H∞(M ) is called inner if F∗(ζ)F(ζ) = I almost everywhere on T (I m,n n n stands for the identity matrix in M ). F is called outer if FH2(Cn) is dense in n,n H2(Cm). Finally, F is called co-outer if the transposed function Ft is outer. An n×n matrix function V, n ≥ 2, is called thematic (see [PY1]) if it is unitary- valued and has the form V = v Θ , where the matrix functions v ∈ H∞(C(cid:0)n) and Θ(cid:1)∈ H∞(M ) are both inner and n,n−1 co-outer. Note that if V is a thematic function, then all minors of V on the first column (i.e., all minors of V of an arbitrary size that involve the first column of V) belong to H∞ ([PY1]). If n = 1 the thematic functions are constant functions whose modulus is equal to 1. Let now Φ ∈ L∞(M ) be a matrix function such that kH k < kH k. It m,n Φ e Φ follows from the results of [PT] that Φ is badly approximable if and only if it admits a representation su 0 Φ = W∗ V∗, 0 Ψ (cid:18) (cid:19) where s > 0, V and Wt are thematic functions, u is a scalar unimodular function (i.e., |u(ζ)| = 1 for almost all ζ ∈ T) such that T is Fredholm, indT > 0, u u and kΨkL∞ ≤ s. Note that in this case s must be equal to kHΦk. In the case Φ ∈ (H∞ +C)(M ) this was proved earlier in [PY1]. m,n Suppose now that kH k is less than the smallest nonzero superoptimal singular Φ e value of Φ and let m ≤ n. It was proved in [PT] that Φ is very badly approximable if and only if Φ admits a thematic factorization, i.e., Φ = W∗W∗···W∗ DV∗ V∗ ···V∗ (2.1) 0 1 m−1 m−1 m−2 0 6 where an m×n matrix function D has the form s u 0 ··· 0 0 ··· 0 0 0 0 s u ··· 0 0 ··· 0 1 1 D =  ... ... ... ... ... ... ... ,  0 0 ··· s u 0 ··· 0   m−1 m−1    u ,··· ,u are unimodular scalar functions such that the operators T are Fred- 0 m−1 uj holm, indT > 0, {s } is a nonincreasing sequence of nonnegative num- uj j 0≤j≤m−1 bers, I 0 I 0 j j W = , V = , 1 ≤ j ≤ m−1, (2.2) j 0 W˘ j 0 V˘ j j (cid:18) (cid:19) (cid:18) (cid:19) and Wt,W˘ t,V ,V˘ are thematic matrix functions, 1 ≤ j ≤ m − 1. Moreover, in 0 j 0 j this case the s are the superoptimal singular values of Φ: s = t , 0 ≤ j ≤ m−1. j j j The indices k of the thematic factorization (2.1) (thematic indices) are defined in j case t 6= 0: k = indT . Note that this result was established earlier in [PY1] in j j uj the case Φ ∈ (H∞ +C)(M ). m,n It also follows from the results of [PT] that if r ≤ min{m,n} is such that t < t , kH k < t and F is a matrix function in Ω , then Φ−F admits a r r−1 Φ e r−1 r−1 factorization t u 0 ··· 0 0 0 0 0 t u ··· 0 0 1 1 Φ−F = W∗···W∗  ... ... ... ... ... V∗ ···V∗, 0 r−1 r−1 0  0 0 ··· t u 0  (2.3)  r−1 r−1   0 0 ··· 0 Ψ      in which the V and W have the form (2.2), the Wt,W˘ t,V ,V˘ are thematic ma- j j 0 j 0 j trix functions, the u are unimodular functions such that T is Fredholm and j uj indTuj > 0, and kΨkL∞ ≤ tr and kHΨk < tr. Factorizations of the form (2.3) with a nonincreasing sequence {t } are j 0≤j≤r−1 called partial thematic factorizations. If Φ − F admits a partial thematic factorization of the form (2.3), then t ,t ,··· ,t are the largest r superoptimal 0 1 r−1 singular values of Φ, and so they do not depend on the choice of a partial thematic factorization. It was observed in [PY1] that the indices of a thematic factorization are not uniquely determined by the matrix function but may depend on the choice of a thematic factorization. On the other hand it follows from the results of [PT] that under the conditions kH k < kH k and t > kH k , the sum of the indices Φ e Φ r−1 Φ e of a (partial) thematic factorization that correspond to all superoptimal singular values equal to a positive specific value is uniquely determined by the function Φ itself. Note that for H∞ +C matrix functions this was proved earlier in [PY2]. 7 In[AP]thenotionofamonotone(partial)thematicfactorizationwasintroduced. It plays a crucial role in this paper. A (partial) thematic factorization is called monotone if for any positive superoptimal singular value t the thematic indices k ,k ,··· ,k thatcorrespondtoallsuperoptimalsingularvaluesequaltotsatisfy r r+1 s k ≥ k ≥ ··· ≥ k . r r+1 s It was shown in [AP] that under the conditions kH k < kH k and Φ e Φ t > kH k , Φ − F admits a monotone partial thematic factorization of the r−1 Φ e form (2.3). Moreover, it was established in [AP] that the indices of a monotone partial thematic factorization are uniquely determined by Φ and do not depend on the choice of a partial thematic factorization. 3. Unitary interpolants and Wiener–Hopf indices We study in this section the problem of finding unitary interpolants for a square matrix function. In other words, given Φ ∈ L∞(M ), the problem is to find a n,n unitary-valued function U such that Φ−U ∈ H∞(M ). For such a problem to n,n be solvable, the function Φ must satisfy the obvious necessary condition kHΦk = distL∞ Φ,H∞(Mn,n) ≤ 1. (3.1) However, it is well known that this con(cid:0)dition is not s(cid:1)ufficient even for scalar functions (see [AAK1]). We assume that the matrix function Φ satisfies the following condition kHΦke = distL∞ Φ,(H∞ +C)(Mn,n) < 1. (3.2) andweareinterested inunitaryinter(cid:0)polantsU ofΦsucht(cid:1)hattheToeplitzoperator T is Fredholm. We show that under conditions (3.1) and (3.2) the function Φ has U such a unitary interpolant. Clearly, (3.2) is satisfied if Φ ∈ (H∞ +C)(M ). n,n As we have already observed in the introduction, a unitary interpolant U of a matrix function Φ satisfying (3.2) is the symbol of a Fredholm Toeplitz operator if and only if kHU∗ke < 1. (3.3) We describe all possible indices of Wiener–Hopf factorizations of such unitary interpolantsofΦintermsofsuperoptimalsingularvaluesandindicesofamonotone partial thematic factorization of Φ−Q, where Q is a best approximation of Φ by bounded analytic matrix functions. Let us first state the results. As we have explained in §2, if Φ satisfies conditions (3.1) and (3.2) and Q ∈ H∞(M ) is a best approximation of Φ by bounded ana- n,n lyticmatrixfunctions, thenΦ−Qadmitsamonotonepartialthematicfactorization 8 of the form u 0 ··· 0 0 0 0 u ··· 0 0 1 Φ−Q = W∗···W∗  ... ... ... ... ... V∗ ···V∗, 0 r−1 r−1 0  0 0 ··· u 0   r−1   0 0 ··· 0 Ψ      where kΨkL∞ ≤ 1 and kHΨk < 1 (here r is the number of superoptimal singular valuesofΦequalto1; itmaycertainlyhappenthatr = 0inwhichcaseΨ = Φ−Q). We denote by k , 0 ≤ j ≤ r −1, the thematic indices of the above factorization. j Theorem 3.1. Let Φ be a matrix function in L∞(M ) such that kH k < 1. n,n Φ e Then Φ has a unitary interpolant U satisfying kHU∗ke < 1 if and only if kHΦk ≤ 1. If U is a unitary interpolant of a matrix function Φ and conditions (3.1), (3.2), and (3.3) hold, we denote by d , 0 ≤ j ≤ n − 1, the Wiener–Hopf factorization j indices of U arranged in the nondecreasing order: d ≤ d ≤ ··· ≤ d . Recall 0 1 n−1 that the indices d are uniquely determined by the function U. j Theorem 3.2. Let Φ be a matrix function in L∞(M ) such that kH k ≤ 1 n,n Φ and kH k < 1. Let r be the number of superoptimal singular values of Φ equal Φ e to 1. Then each unitary interpolant U of Φ satisfying kHU∗ke < 1 has precisely r negative Wiener–Hopf indices. Moreover, d = −k , 0 ≤ j ≤ r −1. j j In particular, Theorem 3.2 says that the negative Wiener–Hopf indices of a unitary interpolant U of Φ that satisfies (3.3) are uniquely determined by Φ. Theorem 3.3. Let Φ and r satisfy the hypotheses of Theorem 3.2. Then for any sequence of integers {d } satisfying j r≤j≤n−1 0 ≤ d ≤ d ≤ ··· ≤ d r r+1 n−1 there exists a unitary interpolant U of Φ such that kHU∗ke < 1 and the nonnegative Wiener–Hopf factorization indices of U are d ,d ,··· ,d . r r+1 n−1 Note that Theorem 3.1 follows immediately from Theorem 3.3. Theorem 3.4. Let Φ satisfy the hypotheses of Theorem 3.2. Then Φ has a unique unitary interpolant if and only if all superoptimal singular values of Φ are equal to 1. Proof of Theorem 3.2. Suppose that U is a unitary interpolant of Φ that satisfies (3.3). Put G = U −Φ ∈ H∞(M ). n,n Let κ ≥ 0. Let us show that if f ∈ H2(Cn), then kH fk = kfk if and only zκΦ 2 2 if f ∈ KerT . zκU Indeed, if f ∈ KerT , then zκUf ∈ H2(Cn). It follows that zκU − kH fk = kH fk = kP (zκUf)k = kzκUfk = kfk . zκΦ 2 zκ(Φ+G) 2 − 2 2 2 9 Conversely, suppose that kH fk = kfk . Then kH fk = kfk . Hence, zκΦ 2 2 zκU 2 2 zκUf ∈ H2(Cn), and so f ∈ KerT . − zκU It follows from formula (1.2) that dimKerT = −d −κ. zκU j {j∈[0,n−X1]:−dj>κ} By Theorem 4.8 of [AP], dim{f ∈ H2(Cn) : kH fk = t kfk } = k −κ. zκΦ 2 0 2 j {j∈[0,rX−1]:kj>κ} Hence, −d −κ = k −κ, κ ≥ 0. (3.4) j j {j∈[0,n−X1]:−dj>κ} {j∈[0,rX−1]:kj>κ} It is easy to see from (3.4) that U has r negative Wiener–Hopf factorization indices and d = −k for 0 ≤ j ≤ r −1. (cid:4) j j To prove Theorem 3.3, we need two lemmas. Lemma 3.5. Let Ψ ∈ L∞(M ) and kH k < 1. Then for any nonnegative m,m Ψ integers d , 0 ≤ j ≤ m−1, there exists a unitary interpolant U of Ψ that admits j a representation u 0 ··· 0 0 0 u ··· 0 1 U = W0∗···Wm∗−1 ... ... ... ... Vm∗−1···V0∗, (3.5)  0 0 ··· u   m−1    where I 0 I 0 j j W = , V = , 1 ≤ j ≤ m−1, j 0 W˘ j 0 V˘ j j (cid:18) (cid:19) (cid:18) (cid:19) V , Wt, V˘ , W˘ t are thematic matrix functions, and u ,··· ,u are unimodular 0 0 j j 0 m−1 functions such that the Toeplitz operators T are Fredholm and uj indT = −d , 0 ≤ j ≤ m−1. uj j Proof of Lemma 3.5. We argue by induction on m. Assume first that m = 1. Letψ beascalarfunctioninL∞ suchthatkH k < 1. Without lossofgenerality we ψ mayassume thatkψk < 1. Consider thefunctionz¯d0+1ψ. Clearly, kH k < 1. ∞ z¯d0+1ψ It is easy to see that there exists c ∈ R such that kH k = 1. z¯d0+1ψ+cz¯ Since H has finite rank, it is easy to see that cz¯ kH k = kH k < 1 = kH k. z¯d0+1ψ+cz¯ e z¯d0+1ψ e z¯d0+1ψ+cz¯ 10

Unitary interpolants and factorization indices of matrix functions

Similar Unitary interpolants and factorization indices of matrix functions