The spectral Carathéodory-Fejér problem

Transcription

1 The spectral Carathéodory-Feér problem H.-N. Huang, S.A.M. Marcantognini and N.J. Young Abstract. The problem of the title is to construct an analytic k k matrixvalued function in the unit disc with a number of prescribed derivatives at 0 and with spectral radius bounded by 1. We show that the problem can be reduced to an interpolation problem for the symmetrized polydisc G k, and thereby show that, in the case of derivatives of orders 0 and 1 being prescribed, the problem is equivalent to the infinitesimal Kobayashi extremal problem for G k, which is solved completely in the case k = 2. Mathematics Subect Classification (2000). Primary 30E05, 47A56; Secondary????? Keywords. spectral Carathéodory-Feér interpolation, Schur functions, spectral Nevanlinna-Pick interpolation theory, symmetrized polydisc, Kobayashi metric. 1. Introduction A celebrated problem in complex analysis from early in the last century is the following [10, 11, 20]: (CF) Given c 0, c 1,..., c n C, determine whether there exist c n+1, c n+2,... C such that the function F (λ) = c λ (1.1) is analytic and bounded by 1 in the open unit disc D. This problem has an elegant solution; it also has relevance to some engineering questions [12, 13], and in consequence its numerous generalizations have been the The first named author s work was done while he was visiting the University of Newcastle upon Tyne and sponsored by National Science Council grant NSC 42153F. The second named author s work was done while she was visiting the University of Newcastle upon Tyne and supported by EPSRC grant GR/S77448/01.

2 2 Huang, Marcantognini and Young subect of many papers. It transpires that a function F with the stated properties exists if and only if the (n + 1) square Toeplitz matrix c c 1 c c 2 c 1 c c n c n 1 c n 2 c 0 is a contraction. Of the many approaches to the proof of this and closely related results we mention the seminal ones [1, 25, 27]. All of these methods extend readily to the matricial and operatorial versions of the problem, in which the c are matrices or operators and the function F is to be chosen so that F (λ) 1 for all λ D, where denotes the operator norm. Some variants of these classical problems have arisen since about 1980 in the context of control engineering [2]. The design of automatic controllers for linear systems necessitates the construction of analytic matrix-valued functions satisfying interpolation conditions and some subtle boundedness conditions. A general formulation, the µ-synthesis problem, has been popularised by J. Doyle [16]. If this general interpolation problem were to find a satisfactory analytic solution it would have considerable significance for linear control theory; for the present, no such solution is in sight, and a number of mathematicians have therefore attempted to analyse special cases of it, such as the spectral Nevanlinna-Pick problem [3, 4, 7, 8]. In this paper we address another special case of the µ-synthesis problem. We vary the motivating problem (CF) above by taking the prescribed Taylor coefficients c 0, c 1,..., c n to be complex k k matrices and asking for the existence of an analytic function F as in equation (1.1) with spectral radius bounded by 1 in D. This apparently modest modification makes the problem much harder. We call the following the spectral Carathéodory-Feér problem: (SCF) Given k k complex matrices V 0, V 1,..., V n, determine whether there exist matrices V n+1, V n+2,... such that the function is analytic in D and satisfies F (λ) = V λ! r(f (λ)) < 1 (1.2) for all λ D, where r( ) denotes the spectral radius of a matrix. The condition that F is given by equation (1.2) can also be stated in the form F () (0) = V, 0 n. If the problem is addressed by an application of the spectral version of the Commutant Lifting Theorem by Bercovici, Foiaş and Tannenbaum [7, 8], a bounded analytic function F with Taylor expansion of the form (1.2) and such

3 The spectral Carathéodory-Feér problem 3 that sup{r(f (λ)) : λ D} < 1 is found to exist if and only if there exist k k matrices E 0,, E n such that E 0 is invertible and ẼṼ Ẽ 1 < 1 with E V E 1 E 0 0 Ẽ =......, Ṽ = V 1 V E n E n 1 E 0 n! V 1 n (n 1)! V n 1 V 0 Our main result (Theorem 2.1) is that if V 0 is a non-derogatory matrix then Problem SCF can be reduced to a question about the complex geometry of a certain domain in C k, the symmetrised polydisc G k. In dimensions 2 and 3 this domain is given by G 2 = {(z 1 + z 2, z 1 z 2 ) : z 1, z 2 D}, G 3 = {(z 1 + z 2 + z 3, z 2 z 3 + z 1 z 3 + z 1 z 2, z 1 z 2 z 3 ) : z 1, z 2, z 3 D}. More generally, if c k m denotes the elementary symmetric polynomial of degree m in k variables, we define G k = { (c k 1(z), c k 2(z),..., c k k(z)) : z D k}. Recall that a square matrix T is said to be non-derogatory if the rational canonical form of T consists of a single block in companion form. Equivalently, T is non-derogatory if it has a cyclic vector, or if all the Jordan cells in the Jordan form of T have distinct eigenvalues. If T is 2 2 then T is non-derogatory if and only if T is not a scalar multiple of the identity matrix. The main result enables us to give a full solution of Problem SCF when k = 2 and n = 1. Theorem 1.1. Let V 0, V 1 be 2 2 complex matrices and suppose that V 0 is nonderogatory. There exists an analytic matrix function F in D such that F (0) = V 0, F (0) = V 1 and r(f (λ)) < 1 for all λ D if and only if where sup s 1 (1 ω 2 p 0 ) p 1 ω(2 ωs 0 ) ω 2 (s 0 s 0 p 0 ) 2ω(1 p 0 2 ) + s 0 p 0 s 0 1, ω =1 s 0 = tr V 0, p 0 = det V 0, (1.3) s 1 = tr V 1, p 1 = v0 11 v12 1 v21 0 v v11 1 v12 0 v21 1 v22 0 (1.4) and V m = [ ] vi m 2, for m = 0, 1. i,=1 To state the result for higher values of n we require some notation for higherorder directional derivatives. Consider an n times Fréchet-differentiable function

4 4 Huang, Marcantognini and Young g : Ω X Y where X, Y are complex Banach spaces and Ω is an open subset of X. We define the directional derivatives n g : Ω X n C inductively for n N by 1 g(x 0, x 1 ) = d dz g(x 0 + zx 1 ) = Dg(x 0 )(x 1 ), z=0 where Dg(x 0 ) is the Fréchet derivative of g at x 0, and n+1 g(x 0, x 1,, x n+1 ) = d dz ng(x 0 + zx 1, x 1 + zx 2,, x + zx n+1 ). z=0 Thus, if we use the notation d/du for the directional derivative along the vector u, we can write n g(x 0, x 1,, x n ) = d d d(x 1, x 2,..., x n ) d(x 1, x 2,..., x n 1 ) d g(x 0 ). dx 1 These directional derivatives can be expressed in terms of higher-order Fréchet derivatives with the aid of the formula named after St. Francis Faà di Bruno 1 : see Note 3 following Theorem 2.1. We apply to the coefficients of the characteristic polynomial of a matrix. For A in the space C k k of k k complex matrices define c m (A), for m = 0, 1,..., k 1, by the equation det(zi A) = z k c k 1 (A)z k ( 1) k c 0 (A) for all z C. Thus c k m is a polynomial of degree m in the entries of A; to be precise, it is the sum of all principal m m minors of A. Let c = (c k 1,..., c 0 ), so that c is a polynomial mapping from C k k to C k. For any non-negative integer m, m c is a polynomial mapping from ( C k k) m+1 to C k. We shall need the following form of the chain rule for higher derivatives. Lemma 1.2. Let F : D C k k be analytic and let g : C k k C k be a polynomial mapping. For any λ D and positive integer m, ( ) (g F ) m (λ) = m g F (λ), F (λ),..., F (m) (λ). (1.5) 1 A remarkable man who not only wrote an influential treatise on binary forms but was also canonized for his charitable and religious works [19].

5 The spectral Carathéodory-Feér problem 5 Proof. When m = 1, this is the usual chain rule. Suppose it is true for m N. Then (g F ) m+1 (λ) = dm dλ m (g F ) (λ) = dm dλ m g (F (λ), F (λ)) = m λ m t g (F (λ) + tf (λ)) t=0 = m t λ m g (F (λ) + tf (λ)) t=0 = ( t mg F (λ) + tf (λ), F (λ) + tf (λ),..., F (m) (λ) + tf (λ)) (m+1) ( ) = m+1 g F (λ), F (λ),..., F (m+1) (λ). t=0 Hence, by induction, the formula holds for all m N. Of course this chain rule holds in much greater generality than we have stated. It does not appear in the admirable scholarly article on higher chain rules by W. P. Johnson [21]. Beside this section, which serves as an introduction, the paper is organized in 5 sections. In 2 we describe how the spectral Carathéodory-Feér problem can be reduced to an interpolation problem for analytic functions from D to G k. The proof of Theorem 1.1 is given in 3. It is obtained by combining results of [6] on the hyperbolic geometry of the symmetrized bidisc G 2 with a result, also established in Section 3, which states that, for prescribed derivatives of order 0 and 1 and, in general, for any k 2, the spectral Carathéodory-Feér problem reduces to the infinitesimal Kobayashi extremal problem for G k. The reduction also yields a necessary condition for solvability. This condition is presented in 3. In 4 we adopt a different approach which leads us to a necessary and sufficient condition for 2 2 matrix functions with any number of prescribed derivatives (the case n 1 and k = 2), but with the disadvantage that the condition involves a nontrivial search process over a (possibly) high-dimensional set. We conclude in 5, where the method of construction presented in Theorem 2.1 is carried out for a particular example. 2. Interpolation into C k k and G k In attempting to construct an analytic C k k -valued function F on D satisfying interpolation conditions and a bound on the eigenvalues of F (λ), one s first thought might be to reduce the target matrices to Jordan form and interpolate the eigenvalues. Bercovici, Foias and Tannenbaum [7] showed the limitations of this approach to the spectral Nevanlinna-Pick problem, and Agler and Young [3, 5] were

6 6 Huang, Marcantognini and Young therefore motivated to develop an alternative approach based on interpolating the characteristic polynomial. If the desired function F is to satisfy F (λ ) = W, 1 n, then c m F is an analytic function in D that maps λ to c m (W ), for m = 0, 1,..., k 1, and the condition that r(f (λ)) < 1 for λ D translates into the requirement that (c k 1 F,..., c 0 F ) is an analytic function from D to G k. Accordingly it is natural to study analytic interpolation from D to G k, and this idea has led to some progress on special cases of the spectral Nevanlinna-Pick problem [3, 4, 5, 15, 26]. In this section we show that, subect to a genericity condition, the spectral Carathéodory-Feér problem can also be reduced to the problem of interpolation by analytic functions from D to G k. For any non-negative integer n we define the interpolation body I n (0, G k ) to be the set {( } I n (0, G k ) = h(0), h (0),..., h (0)) (n) h : D Gk is analytic. Thus I n (0, G k ) C (n+1)k. Theorem 2.1. Let V 0, V 1,..., V n C k k and suppose that V 0 is non-derogatory. Let w 0 = c(v 0 ), w m = m c(v 0, V 1,..., V m ), for m = 1, 2,..., n. There exists an analytic function F : D C k k such that and F (m) (0) = V m, r(f (λ)) < 1 m = 0, 1,..., n for all λ D if and only if (w 0, w 1,..., w n ) I n (0, G k ). Moreover, if these equivalent conditions are satisfied then there is a bounded solution F of the interpolation problem with the property that F (λ) is non-derogatory for every λ D. By way of orientation, let us calculate a few w c = (tr, det) and so, if we write in the case k = 2. Here e 1 = [1 0] T, e 2 = [0 1] T,

7 The spectral Carathéodory-Feér problem 7 then we have w 0 = (tr V 0, det V 0 ), (2.1) d w 1 = (tr V 0, det V 0 ) dv 1 = d dt (tr(v 0 + tv 1 ), det(v 0 + tv 1 )) t=0 = (tr V 1, det [V 0 e 1 V 1 e 2 ] + det [V 1 e 1 V 0 e 2 ]), (2.2) w 2 = d d(v 1, V 2 ) (tr V 1, det [V 0 e 1 V 1 e 2 ] + det [V 1 e 1 V 0 e 2 ]) = (tr V 2, det [V 0 e 1 V 2 e 2 ] + 2 det V 1 + det [V 2 e 1 V 0 e 2 ]). (2.3) Proof of necessity. Suppose there exists an F satisfying the stated conditions. Let h = c F, so that h is analytic from D to C k. For any λ D the characteristic polynomial of F (λ) is z k c k 1 F (λ)z k ( 1) k c 0 F (λ), and since r(f (λ)) < 1, the zeros of this polynomial lie in D. It follows that Hence h(d) G k, and so By Lemma 1.2, for m = 0, 1,..., n, Hence (c k 1 F (λ),..., c 0 F (λ)) G k. (h(0), h (0),..., h (n) (0)) I n (0, G k ). h (m) (0) = (c F ) (m) (0) = m c(f (0),..., F (m) (0)) = m c(v 0,..., V m ) = w m. (w 0, w 1,..., w n ) I n (0, G k ). Before embarking on the proof of sufficiency let us recall a little of the theory of rational canonical forms of matrices, e.g. [23]. Let L denote the left shift operator on the space C[[z]] of formal power series: if x(z) = n=0 x nz n then Lx(z) = x 1 + x 2 z + x 3 z 2 + = 1 (x(z) x(0)). z Consider a monic polynomial f of degree k, f(z) = z k + a k 1 z k a 1 z + a 0. The L-invariant subspace Ker f(l) of C[[z]] can be described as the space of x C[[z]] expressible in the form x = u/ f

8 8 Huang, Marcantognini and Young for some u in the space P k 1 of polynomials of degree less than k, where f(z) = z k f(1/z) = 1 + a k 1 z + + a 1 z k 1 + a 0 z k. Note that L Ker f(l) has minimal polynomial equal to its characteristic polynomial f, and so is a non-derogatory linear transformation. The theory of the rational canonical form tells us that all non-derogatory matrices with the same characteristic polynomial are similar; thus the general non-derogatory matrix with characteristic polynomial f can be obtained by taking the matrix of L Ker f(l) with respect to a suitable basis of Ker f(l). Consequently, to construct a matrix F (λ) with known characteristic polynomial f(, λ), a possible method is to choose a basis (depending on λ) of Ker f(l, λ) and to define F (λ) to be the matrix of L Ker f(l, λ) with respect to this basis. The desired properties of F (λ) can be converted into requirements of the basis, and so the problem is to construct, for each λ, a basis u 1 (, λ),..., u k (, λ) of P k 1 such that the basis u (, λ) / f(, λ), 1 k, of Ker f(l, λ) satisfies suitable conditions. To say that F (λ) = [F i (λ)] is the matrix of the restriction of L with respect / to the u i f is the statement that, for all z C, λ D and 1 k, where 1 z ( u (z, λ) f(z, λ) u (0, λ) f(0, λ) ) = i=1 f(z, λ) = z k f(1/z, λ); that is, since f(, λ) is monic and f(0, λ) = 1, u (z, λ) f(z, λ)u (0, λ) = z F i (λ) u i(z, λ) f(z, λ) (2.4) F i (λ)u i (z, λ). (2.5) For the present purpose we wish F and its derivatives to take prescribed values at 0. By differentiating the relation (2.5) repeately and substituting λ = 0 we obtain relations which the u must satisfy if F is to have the desired properties. We shall show that the u can be chosen to satisfy these relations, and that the resulting F will indeed be the desired interpolating function. Proof of sufficiency. Suppose that i=1 (w 0, w 1,..., w n ) I n (0, G k ). That is, there exists an analytic function h : D G k such that which is to say that for m = 1, 2,..., n. (h(0), h (0),..., h (n) (0)) = (w 0, w 1,..., w n ) h(0) = c(v 0 ) = (c k 1 (V 0 ),..., c 0 (V 0 )), h (m) (0) = w m = m c(v 0, V 1,..., V m ) (2.6) = ( m c k 1 (V 0,..., V m ),..., m c 0 (V 0,..., V m ))

9 The spectral Carathéodory-Feér problem 9 In fact we may choose h to be a rational function (e.g.[15]), which implies that h is analytic in a domain Ω that contains the closed unit disc; we do not claim that h(ω) G k. For λ Ω define the monic polynomial f(, λ) of degree k by f(z, λ) = z k h 1 (λ)z k ( 1) k h k (λ) (2.7) where h = (h 1,..., h k ). The statement that h(λ) G k means precisely that the zeros of the polynomial f(, λ) lie in D. Hence, if we define a matrix F (λ) so that its characteristic polynomial is f(, λ) then we shall have r(f (λ)) < 1 for all λ D. Let q be the characteristic polynomial of V 0 : q(z) = z k c k 1 (V 0 )z k ( 1) k c 0 (V 0 ) = z k h 1 (0)z k ( 1) k h k (0) = f(z, 0). Define further, for λ Ω, f(z, λ) = z k f(1/z, λ) = 1 h 1 (λ)z + + ( 1) k h k (λ)z k, (2.8) q(z) = z k q(1/z) = 1 c k 1 (V 0 )z + + ( 1) k c 0 (V 0 )z k = f(z, 0), E(λ) = Ker f(l, λ). Thus E(λ) is an L-invariant subspaace of C[[z]] and E(0) = Ker q(l) = {u/ q : u P k 1 }. Since V 0 is non-derogatory there is a basis χ 1 / q,..., χ k / q of E(0) with respect to which the matrix of L E(0) is V0 ; here χ 1,..., χ k is a basis of P k 1, and we have (compare equation (2.5)) χ (z) q(z)χ (0) = z viχ 0 i (z), i where we define V m = [ vi m ] k, i,=1 m = 0, 1,..., n. (2.9) Let us write χ = [χ 1 χ k ] T, so that the defining property of the χ can be expressed Since q(z) = det(1 zv T 0 ) we have (1 zv T 0 )χ(z) = q(z)χ(0). (2.10) χ(z) = q(z)(1 zv T 0 ) 1 χ(0) = ad(1 zv T 0 )χ(0). (2.11) We denote by P k 1 C k k the space of k k matricial polynomials with each entry of degree less than k, and we define a sequence A m : ( C k k) m+1 Pk 1 C k k, m = 0, 1,..., n,

10 10 Huang, Marcantognini and Young of polynomial mappings inductively as follows. For X 0, X 1,..., X n C k k, A 0 (X 0 ) is the polynomial matrix and for m = 0, 1,..., n 1, A 0 (X 0 ) = ad(1 zx 0 ), A m+1 (X 0,..., X m+1 ) = d d(x 1,..., X m+1 ) Am (X 0,..., X m ) = d dt Am (X 0 + tx 1,..., X m + tx m+1 ). (2.12) t=0 Observe that while, for m 0, A 0 (X 0 )(0) = ad 1 = 1 A m+1 (X 0,..., X m+1 )(0) = d d(x 1,..., X m+1 ) Am (X 0,..., X m )(0) and we find, by induction, that for m 1, identically in X 0,..., X m. The A m satisfy a crucial recursive relation. A m (X 0,..., X m )(0) = 0 (2.13) Lemma 2.2. For m 1, z C and k k matrices X 0, X 1, X 2..., { A m (X 0,..., X m ) = m c(x 0, X 1,..., X m )Z where + z m 1 ( m } )A (X 0,..., X )X m (1 zx 0 ) 1 (2.14) Z = [ z z 2 ( z) k ] T. Proof. Since det(1 zx 0 ) = 1 + c(x 0 )Z we have By definition, A 1 (X 0, X 1 ) = d det(1 zx 0 ) = d c(x 0 )Z = c(x 0, X 1 )Z. dx 1 dx 1 = d A 0 (X 0 ) = d ad(1 zx 0 ) dx 1 dx 1 d { det(1 zx 0 )(1 zx 0 ) 1} dx 1 = c(x 0, X 1 )Z(1 zx 0 ) 1 + det(1 zx 0 ) d dx 1 (1 zx 0 ) 1.

11 The spectral Carathéodory-Feér problem 11 Now d dx 1 (1 zx 0 ) 1 = d dt (1 z(x 0 + tx 1 )) 1 t=0 = (1 zx 0 ) 1 zx 1 (1 zx 0 ) 1 so that A 1 (X 0, X 1 ) = = } { c(x 0, X 1 )Z + det(1 zx 0 )(1 zx 0 ) 1 zx 1 (1 zx 0 ) 1 } { c(x 0, X 1 )Z + za 0 (X 0 )X 1 (1 zx 0 ) 1, which is equation (2.14) with m = 1. Suppose equation (2.14) holds for m up to and including a particular m 1. Then A m+1 (X 0,..., X m+1 ) = d d(x 1,..., X m+1 ) Am (X 0,..., X m ) = d d(x 1,..., X m+1 )( { m c(x 0,..., X m )Z = +z m 1 ( ) m )A (X 0,..., X )X m }(1 zx 0 ) 1 { m+1 c(x 0,..., X m+1 )Z +z m 1 ( ) m d ( A ) } (X 0,..., X )X m (1 zx 0 ) 1 d(x 1,..., X m+1 ) { + m c(x 0,..., X m )Z + z m 1 ( m } )A (X 0,..., X )X m d d(x 1,..., X m+1 ) (1 zx 0) 1. (2.15) The second term on the right hand side of equation (2.15) is { m c }(1 zx 0 ) 1 zx 1 (1 zx 0 ) 1 = A m (X 0,..., X m )zx 1 (1 zx 0 ) 1,

12 12 Huang, Marcantognini and Young while m 1 = ( m ) d d(x 1,..., X m+1 ) m 1 ( m = A 0 (X 0 )X m+1 + = ( m + m 1 m ( m + 1 ( A (X 0,..., X )X m ) ) A +1 (X 0,..., X +1 )X m + m 1 =1 { ( ) m + 1 ) A m (X 0,..., X m )X 1 ( m m 1 ( ) m A (X 0,..., X )X m +1 ) } A (X 0,..., X )X m+1 ) A (X 0,..., X )X m+1 A m (X 0,..., X m )X 1. These relations, together with equation (2.15) yield { A m+1 (X 0,..., X m+1 ) = m+1 c(x 0, X 1,..., X m+1 )Z + m ( m + 1 } z )A (X 0,..., X )X m+1 (1 zx 0 ) 1, which is equation (2.14) with m replaced by m + 1. By induction, equation (2.14) holds for all m N. Now define χ [m] = A m (V 0,..., V m ) T χ(0), m = 0, 1,..., n. (2.16) Each χ [m] lies in the space P k 1 C k of column vectors whose entries belong to P k 1, and in view of the relation (2.11) we have χ [0] = χ, while by virtue of equation (2.13), χ [m] (0) = 0 for m 1. We claim that there exist functions u m (z, λ), m = 1, 2,..., k with the following properties: (U1) u 1 (, λ),..., u k (, λ) is a basis of P k 1 for each λ C; (U2) u (z, ) is entire for z C and = 1, 2,..., k; u 1 (z, λ) m (U3) λ m. = χ [m] (z), m = 0, 1,..., n. u k (z, λ) λ=0 Indeed, since χ 1,..., χ k is a basis of P k 1, there are matrices B 1,..., B n such that Construct a matrix polynomial χ [m] = B m χ, m = 1, 2,..., n. Q(λ) = Q 1 λ + Q 2 λ Q n λ n

13 The spectral Carathéodory-Feér problem 13 such that the Taylor expansion about 0 of e Q begins e Q(λ) = 1 + B 1 λ + B 2 2! λ2 + + B n n! λn + and define the u by u 1 (z, λ). u k (z, λ) = e Q(λ) χ(z), λ C. (2.17) Since e Q(λ) is nonsingular for any λ C and χ 1,..., χ k is a basis of P k 1, it follows that the components u (, λ) of e Q(λ) χ also constitute a basis of P k 1 for any λ C. It is clear that each u (z, ) is an entire function, and we have m λ m u 1 (z, λ).. = m λ u k (z, λ) m eq(λ) χ(z) λ=0 λ=0 ( = m λ m 1 + B 1 λ + B ) 2 2! λ2 + = B m χ(z) = χ [m] (z) χ(z) λ=0 for m = 0, 1,..., n. Having chosen the u satisfying (U1), (U2) and (U3) we define F (λ) for λ Ω to be the matrix of L Ker f(l, λ) with respect to the basis u 1 (, λ) f(, λ),..., u k(, λ) f(, λ). In particular, F (0) is the matrix of L Ker q(l) with respect to χ 1 / q,..., χ k / q, and so, by choice of the χ, F (0) = V 0. (2.18) For λ Ω (recall the relation (2.5)) which relation can be written where u (z, λ) = f(z, λ)u (0, λ) + z F i (λ)u i (z, λ), i=1 u(z, λ) = f(z, λ)u(0, λ) + zf (λ) T u(z, λ), (2.19) u(z, λ) = [u 1 (z, λ) u k (z, λ)] T. From equation (2.19) we can deduce that F is analytic in Ω. For any λ 0 Ω, u 1 (, λ 0 ),..., u k (, λ 0 ) is a basis of P k 1, and it follows that there is a neighbourhood N of λ 0 and points z 1,..., z k C \ {0} such that det [u i (z, λ)] 0 for all λ N.

14 14 Huang, Marcantognini and Young From the relation u(z, λ) f(z, λ)u(0, λ) = F (λ) T z u(z, λ) for = 1,..., k and λ N and the fact that the matrix [z 1 u(z 1, λ) z k u(z k, λ)] is analytic and nonsingular on N it follows that F is analytic at λ 0. We wish to deduce from equation (2.19) that F (m) (0) = V m for m = 1,..., n. Note first that on differentiating equation (2.8) we have m λ f(z, m λ) = δ m0 h (m) 1 (λ)z + + ( 1) k h (m) k (λ)z k, and so, from equation (2.6), for m 0, m λ f(z, m 0) = δ m0 + ( 1) m c k (V 0,..., V m )z =1 = δ m0 + m c(v 0,..., V m )Z. (2.20) Note also, from property (U3) of the u, that for m 1, and so while and in particular m λ m u(z, 0) = χ[m] (z) = A m (V 0,..., V m )(z) T χ(0) (2.21) m λ m u(0, 0) = χ[m] (0) = 0 u(z, 0) = χ(z) = A 0 (V 0 )(z) T χ(0) (2.22) u(0, 0) = χ(0) 0. Suppose it is true that F () (0) = V for = 1, 2,..., m 1 where 1 m n. Differentiate equation (2.19) m times and put λ = 0 to obtain m m λ m u(z, 0) = ( ) m λ f(z, 0) m u(0, 0) λm m ( m + z )F () (0) T m u(z, 0). λm In the first sum only the = m term is non-zero, and we can write the equation (1 zv0 T ) m m u(z, 0) = λm λ f(z, m 1 ( ) m m 0)χ(0) + z V T m u(z, 0) λm +zf (m) (0) T u(z, 0),

15 The spectral Carathéodory-Feér problem 15 and hence, in view of equations (2.21), (2.22) and (2.20), zf (m) (0) T χ(z) = (1 zv0 T )A m (V 0,..., V m ) T χ(0) m c(v 0,..., V m )Zχ(0) m 1 ( ) m z V T A m (V 0,..., V m ) T χ(0). By Lemma 2.2, (1 zv0 T )A m (V 0,..., V m ) T = m c(v 0,..., V m )Z m 1 ( ) m +z V T A m (V 0,..., V m ) T. On combining the last two equations we find that zf (m) (0) T χ(z) = zv T m A 0 (V 0 ) T χ(0) = zv T m χ(z). As this holds for all z C and span{χ(z) : z C} = C k, it follows that F (m) (0) = V m. Hence, by induction, this relation holds for m = 0, 1,..., n. F is analytic on Ω, a domain containing the closure of D, and hence F is bounded on D. For λ D, h(λ) G k and hence the characteristic polynomial f(z, λ) = z k h 1 (λ)z k ( 1) k h k (λ) of F (λ) has all its zeros in D, whence r(f (λ)) < 1, and so F satisfies the requirements of the theorem. Thus sufficiency is proved. Since F (λ) is the matrix of L Ker f(l, λ) with respect to some basis, F (λ) is non-derogatory for all λ D. Note 1. In the statement of Theorem 2.1 the hypothesis that V 0 be non-derogatory is not redundant. Consider the case that n = 1, k = 2 and V 0 = 0. We ask whether there is an analytic matrix function F in D such that F (0) = 0, F (0) = V 1 and r(f (λ)) < 1 for all λ D. (2.23) If such an F exists we may write F (λ) = λg(λ), where G is an analytic function in D satisfying G(0) = V 1 and r(g(λ)) < 1 for all λ D. It follows that r(v 1 ) < 1 and so det V 1 < 1. Choose V 1 such that tr V 1 = 0 and det V 1 = 2. Then the problem (2.23) has no analytic solution F. However Hence w 0 = c(0) = (0, 0), w 1 = d dv 1 c(v 0 ) = d dt (tr tv 1, det tv 1 ) t=0 = (tr V 1, 0) = (0, 0). (w 0, w 1 ) = (0, 0, 0, 0) I 1 (0, G 2 ). Thus, for derogatory V 0, sufficiency can fail in the statement of Theorem 2.1. Necessity remains true, as the proof above shows.

16 16 Huang, Marcantognini and Young Note 2. The arguments in the proof of Theorem 3 can be slightly modified to show that, subect to the genericity condition, there exists an analytic function F : D C k k, with the prescribed derivatives, but satisfying if and only if r(f (λ)) 1 for all λ D, (w 0, w 1,..., w n ) I n (0, Γ k ), where Γ k denotes the closure of G k. Note 3. The vectors w 0,, w n C k that occur in Theorem 2.1 can be written down more explicitly with the aid of Faà di Bruno s formula. For m 1, w m = m c(v 0,, V m ) = m! b 1! b m!(1!) b1 (m!) Dk c(v 0 )(V 1,, V 1,, V bm }{{} m,, V }{{ m ), } b m where the sum is over all different solutions in nonnegative integers b 1,, b m of b 1 + 2b mb m = m, and k := b 1 + b b m. In the above, D k c(v 0 ) stands for the k-th Fréchet derivative of c at V 0. Also, via a multivariate Faà di Bruno formula [14], the r-th coordinate of w m, m 1, can be expressed in terms of the derivatives of the coordinate function c k r of c as Dz λ c k r (V 0 ) m V q (m!) (q!)(!), q where 1 λ m p(m, λ) = { (q 1,, q m ) : p(m,λ) =1 m q = λ, =1 b 1 m q = m }, and, in the standard multivariate notation, if V = [ ] k v i is a k k complex matrix and µ = (µ 11,, µ 1k,, µ k1,, µ kk ) is a k 2 -tuple of nonnegative integers, i,=1 then µ = µ i, µ! = i,=1 k i,=1 (µ i!), =1 D z µ = µ z µ zµ kk, for µ > 0, kk Dz 0 = identity operator, k V µ = (v i ) µi. i,=1 In the above, the vectors q are k 2 -dimensional, and the coordinate functions c k r of c are to be understood as polynomials in the variables z 11,, z 1k,, z k1,, z kk so that D λ z c k r (V 0 ) = D λ z c k r (v 0 11,, v 0 1k,, v 0 k1,, v 0 kk).

17 The spectral Carathéodory-Feér problem The Carathéodory-Feér and Kobayashi problems for G k Theorem 2.1 reduces the spectral Carathéodory-Feér problem to an analogous problem of interpolation by analytic functions from D to G k. Let us name this problem as in the section heading and denote it by (CFG k ): (CFG k ) Given w 0, w 1,..., w n C k determine whether there exists an analytic function h : D G k such that h () (0) = w, = 1, 2,..., n. The virtue of this reduction (SCF to CFG k ) lies in the fact that there are complex-geometric and operator-theoretic methods which yield considerable information about problem CFG k. A number of recent papers [4, 6, 15, 18, 22] address the analysis and geometry of G k, and although many problems remain, there are definitive results in special cases which reveal a rich structure and connections with other areas. For example, when n = 1, the spectral Carathéodory-Feér problem reduces to the infinitesimal Kobayashi extremal problem for G k, a problem which has been solved in the case k = 2, whence follows the full solution of Problem SCF in the case n = 1, k = 2, as given in Theorem 1.1. We shall indicate the closure of G k by Γ k. For domains Ω 1, Ω 2 we denote by O(Ω 1, Ω 2 ) the space of analytic maps from Ω 1 to Ω 2, and for z C k we denote by O(z) the algebra of germs of analytic functions at z. Consider a domain Ω C k and denote the complex tangent space of Ω by T Ω. For z = (z 1,, z k ) Ω and v = (v 1,, v k ) we shall denote by (z; v) the element of T Ω ϕ (z; v) : O(z) C : ϕ v (z). z The Kobayashi pseudometric on Ω is the function k Ω : T Ω R + defined by =1 k Ω (z; v) = inf u over all u C such that there exists h O(D, Ω) for which The relation (3.1) is equivalent to h (0; u) = (z; v). (3.1) h(0) = z and h (0)u = v. Lemma 3.1. For any k N, z G k and v C k the following are equivalent: (1) there exists h O(D, G k ) such that h(0) = z and h (0) = v; (2) k Gk (z; v) 1. Proof. If (1) holds then h (0; 1) = (z; v) and it is immediate from the definition of k Gk that k Gk (z; v) 1. Suppose, to prove the converse, that k Gk (z; v) 1. For every ε > 0 there exist u ε C with u ε 1 and h ε O(D, G k ) such that h ε (0) = z and h ε(0)u ε = v. On passing to convergent subsequences we may infer that there exist u C with u 1 and an analytic function h : D Γ k such that h(0) = z and h (0)u = v.

18 18 Huang, Marcantognini and Young We claim that h(d) G k. Introduce the function ρ : C k R + given by ρ(s 1,, s k ) = r (3.2) s 1 s 2 s 3 ( 1) k s k ρ is continuous and plurisubharmonic on C k, and ρ h is continuous and subharmonic on D. Furthermore G k = {s C k : ρ(s) < 1}, Γ k = {s C k : ρ(s) 1}, and 0 ρ h 1 on D. It follows from continuity and subharmonicity that {z D : ρ h(z) = 1} is open and closed in D, hence is or D. As ρ h(0) < 1 we must have ρ h < 1 on D, and hence h(d) G k as claimed. Now define H(λ) = h(uλ) for λ D. We have H O(D, G k ), H(0) = z and H (0) = uh (0) = v. Hence (2) (1). We observe that Lemma 3.1 remains true if G k is replaced by any taut domain. We can now show the close connection between the problem SCF in the case n = 1 and the infinitesimal Kobayashi extremal problem. Theorem 3.2. Let V 0, V 1 C k k and suppose V 0 is non-derogatory. Let w 0 = c(v 0 ), w 1 = c(v 0, V 1 ). There exists an analytic function F : D C k k such that and if and only if F (0) = V 0, F (0) = V 1 r(f (λ)) < 1 for all λ D k Gk (w 0 ; w 1 ) 1. (3.3) Proof. By Theorem 2.1, the desired F exists if and only if (w 0, w 1 ) I 1 (0, G k ), that is, if and only if there exists an analytic function h : D G k such that h(0) = w 0, h (0) = w 1. By Lemma 3.1, such an h exists if and only if k Gk (w 0 ; w 1 ) 1. The quantities w 0, w 1 in the condition (3.3) are easily calculated: the components of w 0 are the coefficients in the characteristic polynomial of V 0 (with appropriate signs), while w 1 = ( c k 1 (V 0, V 1 ),, c 0 (V 0, V 1 ))

19 The spectral Carathéodory-Feér problem 19 and c k m (V 0, V 1 ) is the sum of all m m determinants obtainable by taking a principal m m submatrix of V 0 and replacing one column by the corresponding entries of V 1. Alternatively, one may write c k m (V 0, V 1 ) = tr d dt m (V 0 + tv 1 ) t=0. A more difficult step is to calculate the Kobayashi metric k Gk (w 0, w 1 ). To date we only have a usable formula in the case k = 2. By combining Theorem 3.2 with the results of [6] we obtain Theorem 1.1. Proof of Theorem 1.1. We are given V 0, V 1 C 2 2 with V 0 non-derogatory. In accordance with Theorem 3.2 we introduce w 0 = c(v 0 ) = (s 0, p 0 ), w 1 = c(v 0, V 1 ) = (s 1, p 1 ), and it may be checked that the (s, p ) are indeed given explicitly by the formulae (1.3). By Theorem 3.2 the required interpolating function F exists if and only if k G2 (w 0 ; w 1 ) 1. We claim that k G2 (s 0, p 0 ; s 1, p 1 ) = sup s 1 (1 ω 2 p 0 ) p 1 ω(2 ωs 0 ) ω 2 (s 0 s 0 p 0 ) 2ω(1 p 0 2 ) + s 0 s 0 p 0 (3.4) ω =1 from which relation Theorem 1.1 will follow. By [6, Corollary 4.4] the right hand side of equation (3.4) is equal to the Carathéodory metric c G2 (s 0, p 0 ; s 1, p 1 ). Here the Carathéodory metric c G2 : T G 2 R + is defined by { } u c G2 (z; v) = sup 1 ζ 2 : h O(G 2, D), h (z; v) = (ζ; u) Consequently, it suffices to show that c G2 = k G2. By [6, Corollary 5.7] the Carathéodory and Kobayashi distances and the Lempert function of G 2 all coincide: C G2 = K G2 = δ G2. The distances C G2, K G2 are the integrated forms of the Carathéodory and Kobayashi metrics c G2, k G2, respectively [22, Remark (a),(c)]. Moreover G 2 is a taut domain [24, p. 476], as one easily shows with the aid of the defining function ρ in equation (3.2). It follows [22, Proposition 1.2.6] that c G2 = k G2, and hence equation (3.4) is established. Theorem 1.1 follows. Note that non-generic case (V 0 derogatory) is easily handled when k = 2 by Schur reduction. If V 0 = αi and F is a solution of Problem SCF then the function F 1 (λ) = 1 (F (λ) αi)(i ᾱf (λ)) 1 λ

20 20 Huang, Marcantognini and Young is the solution to an analogous problem, with one fewer derivative prescribed, and F 1 (0) = (1 α 2 ) 1 V 1. If V 1 is also derogatory one may iterate this process. If F 1 is a solution to the new problem one may easily recover a solution F of the original one. 4. A necessary condition The reduction of the spectral Carathéodory-Feér problem to Problem CFG k also leads us to a necessary condition for solvability. D. Ogle [26] introduced the oneparameter family of functions Φ ω : G k D, ω T, given by Φ ω (s 1, s 2,..., s k ) = kωk 1 s k + (k 1)ω k 2 s k s 1 k + (k 1)ωs ω k 1 s k 1 and showed that Φ ω maps G k analytically to D; see also [15]. Hence, if there is a solution h to Problem CFG k, then for each ω T, Φ ω h is an analytic selfmap of D. By the classical Carathéodory-Feér theorem, the (n + 1)-square lower triangular Toeplitz matrix with first column [ ] T Φ ω h(0) (Φ ω h) (0) (Φ ω h) (n) (0) is a contraction. The entries (Φ ω h) () (0) can be expressed in terms of the data w 0,..., w n of Problem CFG k. If we combine this observation with Theorem 2.1 we obtain a family of inequalities, indexed by ω T, necessary for the solvability of Problem SCF. We state the result in the case n = 1 to give the flavour of the condition. Theorem 4.1. Let V 0, V 1 C k k and let c(v 0 ) = (w 1 0,..., w k 0), c(v 0, V 1 ) = (w 1 1,..., w k 1), w 0 0 = 1. If there exists an analytic function F : D C k k such that F (0) = V 0, F (0) = V 1 and r(f (λ)) < 1 for all λ D then, for any ω T, the 2 2 Toeplitz matrix [ ] A(ω) 0 B(ω) A(ω) is a contraction, where A(ω) = B(ω) = k =1 w 0 ω 1/ k 1 i=1 (k )w 0 ω, 2 / k 1 (i )w 0 wi 1ω i+ 1 (k )w 0 ω.

21 The spectral Carathéodory-Feér problem 21 Proof. Suppose F as described exists. By Theorem 2.1 there exists h = (h 1,..., h k ) O(D, G k ) such that h(0) = (w 1 0,..., w k 0), h (0) = (w 1 1,..., w k 1) We define h 0 (λ) to be 1 for λ D. For ω T, and hence where N = and Thus and = = k Φ ω h = h ω 1/ (k )h ω, i=0 (Φ ω h) = N/D 2 ( ) ( ) (k )h ω ih i ω i 1 h ω 1 (k i)h i ω i h h iω i+ 1 {(k )i (k i)} i, (i )h h iω i+ 1 i, Φ ω h(0) = D = i=1 (k )h ω. =1 = A(ω) w 0 ω 1/k 1 (k )w 0 ω (Φ ω h) (0) = N(0)/D(0) 2 / k 1 = k (i )w 0 wi 1ω i+ 1 (k )w 0 ω = B(ω). Since Φ ω h is an analytic self-map of D, it follows that the Toeplitz matrix [ ] A(ω) 0 i=0 2 is a contraction. B(ω) A(ω)

22 22 Huang, Marcantognini and Young It is natural to ask whether the necessary condition in Theorem 4.1 is also sufficient for the solvability of Problem SCF in the case of non-derogatory V 0. When k = 2 the answer is yes; the contractivity of the Toeplitz matrix reduces precisely to the necessary and sufficient condition given in Theorem 1.1. We do not know whether the converse to Theorem 4.1 holds for k > 2, nor, in the case n > 1, k 2, whether the analogue of Theorem 4.1 gives a sufficient condition. 5. A necessary and sufficient condition There is an alternative approach to spectral interpolation problems for 2 2 matrix functions which has the merit of yielding a necessary and sufficient condition, but with the drawback that to check this condition one must conduct a search over a (possibly) high-dimensional set. In [5, 9] this approach yielded a solvability condition for the spectral Nevanlinna-Pick problem; here we apply it to problem SCF (with k = 2). Theorem 5.1. Let V 0, V 1,..., V n C 2 2 and suppose V 0 is non-derogatory. The following two statements are equivalent. (1) There exists an analytic function F : D C 2 2 such that F () (0) = V, = 0, 1,..., n, and r(f (λ)) 1 for all λ D. (2) There exist b 0, b 1,..., b n, c 0, c 1,..., c n C such that, for 0 m n, m ( ) ( ) m tr Vm tr V m det(v 0,..., V m ) = b m c 4 and the (2n + 2)-square block Toeplitz matrix R R 1 R R n R n 1 R 0 is a contraction, where, for 0 n, [ 1 R = 2 tr V b 1 c 2 tr V ]. (5.1) Proof. ( ) Suppose (1) holds and let s = tr F, p = det F. Certainly s, p are analytic in D, and since r(f (λ)) 1 for λ D, we have (s(λ), p(λ)) Γ 2 and so s, p H. By a theorem of F. Riesz there exist b, c H such that b = c a.e. on T and b(λ)c(λ) = s(λ)2 p(λ), λ D. (5.2) 4 Let [ 1 ] H(λ) = 2s(λ) b(λ) 1 c(λ) 2 s(λ).

23 The spectral Carathéodory-Feér problem 23 Then H belongs to the Schur class [5, 9], and if we let b = b () (0), c = c () (0), = 0, 1,, n, then we have, for 0 n, [ 1 ] [ H () (0) = 2 s() (0) b () (0) 1 c () 1 = 2 tr V ] b (0) 2 s() 1 (0) c 2 tr V = R. By the classical Carathéodory-Feér theorem, the matrix (5.1) is a contraction. Moreover, if we differentiate the relation (5.2) m times, 0 m n, we obtain m ( ) m b (m ) (0)c () (0) = 1 m ( ) m s (m ) (0)s () (0) p (m) (0). 4 Hence, for 0 m n, m det(v 0,..., V m ) = dm det F (λ) dλm = p (m) (0) λ=0 m ( ) { } m 1 = 4 s(m ) (0)s () (0) b (m ) (0)c () (0) m ( ) { } m tr Vm tr V = b m c. 4 Thus (1) (2). ( ) Suppose (2) holds. By the classical Carathéodory-Feér theorem there exists an analytic function H : D C 2 2 such that H () (0) = R, = 0,..., n, and H(λ) 1 for all λ D. Let h = c H, with c = (tr, det). Then h : D Γ 2 is analytic and so (h(0), h (0),..., h (n) (0)) I n (0, Γ 2 ). For 0 m n, set w m = m c(v 0,..., V m ). Then (h(0), h (0),..., h (n) (0)) = (w 0, w 1,..., w n ), since, for m = 0, 1,..., n, tr R m = tr V m and m det(r 0,..., R m ) = (R dm dz m det 0 + zr m) zm m! z=0 R { ( ) 2 = dm 1 dz m tr V 0 + z tr V zm 4 m! tr V m (b 0 + zb zm m! b m )(c 0 + zc m)} zm m! z=0 c m ( ) { } m tr Vm tr V = b m c 4 = m det(v 0,..., V m ). Under the hypothesis that V 0 be non-derogatory, the result outlined in Note 2 following Theorem 2.1 yields a function F satisfying the conditions in (1).

24 24 Huang, Marcantognini and Young 6. An example The main concern of this paper is the existence or otherwise of a solution to a spectral Carathéodory-Feér problem. It is also of interest to find a solution F, when there is one, and our methods provide some pointers to the construction of a suitable F. Suppose we are given data V 0, V 1,..., V n C k k for Problem SCF, with V 0 non-derogatory. We can easily calculate the quantities w 0 = c(v 0 ), w m = m c(v 0,..., V m ), 1 m n introduced in Theorem 2.1. To give effect to the method of construction used in the proof of Theorem 2.1 we have to find an analytic function h : D G k such that h () (0) = w, 0 n; this is the only non-constructive step in the proof. The finding of a suitable h (when it exists) is thoroughly understood only when n = 1 and k = 2. If such an h can be found for a particular example then the rest of the construction can be carried out. Here is an illustration. Example. Let V 0 = [ ] [ 1 7, V 1 = Find an analytic 2 2 matrix function F such that F (0) = V 0, F (0) = V 1 and r(f (λ)) < 1 for all λ D. Here V 0 is non-derogatory and w 0 = c(v 0 ) = (1, 1 4 ), ( w 1 = c(v 0, V 1 ) = 0, 0 ) = (0, 1 4 ). By Theorem 1.1, the desired F exists if and only if 1 sup 4 (2 ω) ω =1 ω ω The supremum on the left hand side is readily found to be 2 3, and so a solution exists. The next step is to find h O(D, G 2 ) such that h(0) = (1, 1 4 ), h (0) = (0, 1 4 ). By inspection we find that h(λ) = (1, 1 (1 λ)) 4 will do, since the zeros of the polynomial f(z, λ) = z 2 z + 1 (1 λ) 4 are 1 2 (1 ± µ) where µ2 = λ, hence lie in D. Now seek a basis χ 1, χ 2 of P 1 such that the matrix of L Ker f(l, 0) with / respect to the basis χ f(, 0) is V0 ; since V 0 is in fact a companion matrix one ].

25 The spectral Carathéodory-Feér problem 25 can choose χ 1 (z) = z, χ 2 (z) = 1, and it is easy to check that the matrix of L Ker f(l, 0) is indeed V0. Next define χ(z) = [χ 1 (z) χ 2 (z)] T = [z 1] T and, in accordance with equation (2.12), write A 1 d (V 0, V 1 ) = ad(1 zv 0 ) = d dv 1 dt ad(1 zv 0 ztv 1 ) [ ] t=0 1 7 = z and (see equation (2.16)) χ [1] (z) = A 1 (V 0, V 1 ) T χ(0) = [2z z] T. We require functions u 1 (z, λ), u 2 (z, λ) such that (U1) u 1 (, λ), u 2 (, λ) is a basis of P 1 for every λ C; (U2) u [ (z, ) is ] an entire function [ ] for [ = 1, 2 and every z C; u1 (z, 0) z u1 ] [ ] (U3) = χ(z) = and λ (z, 0) 2z = χ u 2 (z, 0) 1 [1] (z) =. (z, 0) z Note that u 2 λ χ [1] = B 1 χ where B = and so (see equation (2.17)) we can take [ ] u1 (z, λ) = e B1λ χ(z) = u 2 that is, [ 2 ] [ e 2λ ] [ ] 0 z, (6.1) 1 1 e 2λ 2 1 u 1 (z, λ) = e 2λ z, u 2 (z, λ) = (1 e2λ )z. We define F (λ) to be the matrix of L Ker f(l, λ) with respect to the basis u 1 (, λ) / f(, λ), u2 (, λ) / f(, λ). A little calculation yields [ 1 ] F (λ) = 2 (e2λ 1) e2λ (λ 4)e 2λ e 2λ 1 2 (3 e2λ ) It may be verified that F (0) = V 0, F (0) = V 1 and the characteristic polynomial of F (λ) is f(z, λ), so that r(f (λ)) < 1 for all λ D. For engineering applications a rational solution F may be preferred. A slight modification of the above construction will yield a rational F. In the definition (6.1) of u 1, u 2, simply replace e B1λ by its Taylor polynomial approximation T m (λ) where m is chosen sufficiently large to ensure that T m (λ) is non-singular for all λ D. In the present example it suffices to take m = 5, and so one obtains a rational interpolant F in which each entry has degree 5.

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27 The spectral Carathéodory-Feér problem 27 [19] history/mathematicians/ Faa di Bruno.html. [20] L. Feér, Über weierstrass sche approximation, besonders durch hermitesche interpolation, Mathemitsche Annalen 102 (1930), [21] W.P. Johnson, The curious history of Faà di Bruno s formula, American Mathematical Monthly 109 (2002), [22] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis revisited, [23] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, [24] S.G. Krantz, Function Theory of Several Complex Variables, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, Calif., [25] B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert Space, Akadémiai Kiadó, Budapest, [26] D. J. Ogle, Operator and function theory of the symmetrized polydisc, Doctoral thesis, University of Newcastle upon Tyne, [27] D. Sarason, Generalized interpolation in H, Transactions of the American Mathematical Society 127 (1967), Acknowledgment The first and second named authors gratefully acknowledge the hospitality of the School of Mathematics and Statistics of the University of Newcastle upon Tyne. H.-N. Huang Department of Mathematics Tunghai University Taichung Taiwan nhuang@mail.thu.edu.tw S.A.M. Marcantognini Department of Mathematics Instituto Venezolano de Investigaciones Científicas P.O. Box Caracas 1020A Venezuela smarcant@ivic.ve N.J. Young School of Mathematics and Statistics University of Newcastle upon Tyne NE1 7RU England N.J.Young@ncl.ac.uk