2 1 0 2 n a J Some analysable instances of mu-synthesis 9 ] V N. J. Young C . To Bill Helton, inspiring mathematician and friend h t a Abstract. Idescribeaverifiablecriterion forthesolvability ofthe2×2 m spectral Nevanlinna-Pick problem with two interpolation points, and [ likewise for three other special cases of the µ-synthesis problem. The problem is to construct an analytic 2×2 matrix function F on the 1 unit disc subject to a finite number of interpolation constraints and a v 3 bound on the cost function supλ∈Dµ(F(λ)), where µ is an instance of thestructured singular value. 7 7 Mathematics SubjectClassification (2010). Primary 93D21, 93B36; Sec- 1 ondary 32F45, 30E05, 93B50, 47A57. . 1 Keywords. Robust control, stabilization, analytic interpolation, sym- 0 metrized bidisc, tetrablock, Carath´eodory distance, Lempert function. 2 1 : v i X 1. Introduction r a It is a pleasure to be able to speak at a meeting in San Diego in honour of Bill Helton, through whose early papers, especially [31], I first became interested in applications of operator theory to engineering. I shall discuss a problem of Heltonian character: a hard problem in pure analysis, with immediate applications in control engineering, which can be addressed by operator-theoreticmethods.Furthermore,the mainadvancesI shalldescribe are based on some highly original ideas of Jim Agler, so that San Diego is the ideal place for my talk. The µ-synthesis problem is an interpolation problem for analytic ma- trix functions, a generalization of the classical problems of Nevanlinna-Pick, Carath´eodory-Fej´erandNehari.Thesymbolµdenotesatypeofcostfunction thatgeneralizesthe operatorandH norms,andthe µ-synthesisproblemis ∞ toconstructananalyticmatrixfunctionF ontheunitdiscsatisfyingafinite number of interpolation conditions and such that µ(F(λ)) ≤ 1 for |λ| < 1. TheprecisedefinitionofµisinSection4below,butformostofthepaperwe needonlyafamiliarspecialcaseofµ –thespectralradiusofasquarematrix A, which we denote by r(A). 2 N. J. Young The purpose of this lecture is to present some cases of the µ-synthesis problem that are amenable to analysis. I shall summarize some results that are scattered through a number of papers, mainly by Jim Agler and me but also severalothers of my collaborators,without attempting to survey all the literature on the topic. I shall also say a little about recent results of some specialists in several complex variables which bear on the matter and may lead to progress on other instances of µ-synthesis. Although the cases to be described here are too special to have sig- nificant practical applications, they do throw some light on the µ-synthesis problem.Moreconcretely,theresultsbelowcouldbeusedtoprovidetestdata forexistingnumericalmethodsandtoilluminate thephenomenon(knownto engineers) of the numerical instability of some µ-synthesis problems. Weareinterestedinciteriaforµ-synthesisproblemstobesolvable.Here is an example. We denote by D and T the open unit disc and the unit circle respectively in the complex plane C. Theorem1.1. Let λ ,λ ∈D be distinct points, let W ,W be nonscalar 2×2 1 2 1 2 matrices of spectral radius less than 1 and let s = trW , p = detW for j j j j j =1,2. The following three statements are equivalent: (1) there exists an analytic function F :D→C2 2 such that × F(λ )=W , F(λ )=W 1 1 2 2 and r(F(λ))≤1 for all λ∈D; (2) (s p −s p )ω2+2(p −p )ω+s −s λ −λ 2 1 1 2 2 1 1 2 1 2 max ≤ ; ω∈T (cid:12)(s1−s¯2p1)ω2−2(1−p1p¯2)ω+s¯2−s1p¯2(cid:12) (cid:12)1−λ¯2λ1(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (3) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 2(cid:12) (cid:12) (2−ωs )(2−ωs )−(2ωp −s )(2ωp −s ) i j i i j j ≥0 " 1−λ¯iλj # i,j=1 for all ω ∈T. The paper is organised as follows. Section 2 contains the definition of the spectral Nevanlinna-Pick problem, sketches the ideas that led to Theo- rem1.1 – reductionto the complex geommetryof the symmetrizedbidisc G, theassociated“magicfunctions”Φ andthecalculationoftheCarath´eodory ω distance on G – and fills in the final details of the proof of Theorem 1.1 us- ing the results of [11]. It also discusses ill-conditioning and the possibility of generalization of Theorem 1.1. In Section 3 there is an analogous solvability criterion for a variant of the spectral Nevanlinna-Pick problem in which the twointerpolationpointscoalesce(Theorem3.1).InSection4,besidesthedef- initionofµandµ-synthesis,thereissomemotivationandhistory.Important workby H.Bercovici,C.Foia¸sandA.Tannenbaumis briefly described,asis Bill Helton’s alternative approach to robust stabilization problems. In Sec- tion 5 we consider an instance of µ-synthesis other than the spectral radius. Analysable instances of µ-synthesis 3 Here we can only obtain a solvability criterion in two very special circum- stances(Theorems5.1and5.2).Thepaperconcludeswithsomespeculations in Section 6. We shall denote the closed unit disc in the complex plane by ∆. 2. The spectral Nevanlinna-Pick problem A particularly appealing special case of the µ-synthesis problem is the spec- tral Nevanlinna-Pick problem: Problem SNP Given distinct points λ ,...,λ ∈ D and k × k matrices 1 n W ,...,W , construct an analytic k×k matrix function F on D such that 1 n F(λ )=W for j =1,...,n (2.1) j j and r(F(λ))≤1 for all λ∈D. (2.2) When k = 1 this is just the classical Nevanlinna-Pick problem, and it is well known that a suitable F exists if and only if a certain n×n matrix formed from the λ and W is positive (this is Pick’s Theorem). We should j j very much like to have a similarly elegant solvability criterion for the case that k > 1, but strenuous efforts by numerous mathematicians over three decades have failed to find one. About 15 years ago Jim Agler and I devised a new approach to the problem in the case k = 2 based on operator theory and a dash of several complex variables ([5] to [13]). Since interpolation of the eigenvalues fails, how about interpolation of the coefficients of the characteristic polynomials of the W , or in other words of the elementary symmetric functions of the j eigenvalues? This thought brought us to the study of the complex geometry ofa certainsetΓ⊂C2, defined below. By this route we were able to analyse quite fully the simplest then-unsolved case of the spectral Nevanlinna-Pick problem: the case n=k =2. For the purpose of engineering application this is a modest achievement,but it nevertheless constituted progress.It had the merit of revealing some unsuspected intricacies of the problem, and may yet lead to further discoveries. 2.1. The symmetrized bidisc Γ We introduce the notation Γ={(z+w,zw):z,w∈∆}, (2.3) G={(z+w,zw):z,w∈D}. 4 N. J. Young ΓandGarecalledtheclosedandopensymmetrizedbidiscsrespectively.Their importance lies in their relation to the sets Σd=ef {A∈C2×2 :r(A)≤1}, Σo d=ef {A∈C2×2 :r(A)<1}. Σ and its interior Σo are sometines called “spectral unit balls”, though the terminology is misleading since they are not remotely ball-like, being un- bounded and non-convex. Observe that, for a 2×2 matrix A, A∈Σ⇔ the zeros of the polynomial λ2−trAλ+detA lie in ∆ ⇔trA=z+w, detA=zw for some z, w ∈∆. We thus have the following simple assertion. Proposition 2.1. For any A∈C2 2 × A∈Σ if and only if (trA,detA)∈Γ, A∈Σo if and only if (trA,detA)∈G. Consequently, if F :D→Σ is analytic and satisfies the equations (2.1) above, where k = 2, then h d=ef (trF,detF) is an analytic map from D to Γ satisfying the interpolation conditions h(λ )=(trW ,detW ) for j =1,...,n. (2.4) j j j Letus assume thatnone ofthe targetmatricesW is a scalarmultiple ofthe j identity.Onthishypothesisitissimpletoshowtheconverse[16]bysimilarity transformation of the W to companion form. j Proposition2.2. Let λ ,...,λ be distinct points in D and let W ,...,W be 1 n 1 n nonscalar 2×2 matrices. There exists an analytic map F : D → C2 2 such × that equations (2.1) and (2.2) hold if and only if there exists an analytic map h:D→Γ that satisfies the conditions (2.4). We have therefore (in the case k =2) reduced the given analytic inter- polation problem for Σ-valued functions to one for Γ-valued functions (the assumptionontheW isharmless,sinceanyconstraintforwhichW isscalar j j may be removed by the standard process of Schur reduction). Why is it an advance to replace Σ by Γ? For one thing, of the two sets, the geometry of Γ is considerably the less rebarbative. Σ is an unbounded, non-smooth4-complex-dimensionalset with spikes shooting off to infinity in many directions. Γ is somewhat better: it is compact and only 2-complex- dimensional, though Γ too is non-convex and not smoothly bounded. But the true reason that Γ is amenable to analysis is that there is a 1-parameter familyoflinearfractionalfunctions,analyticonG,thathasspecialproperties vis-a`-vis Γ. For ω in the unit circle T we define 2ωp−s Φ (s,p)= . (2.5) ω 2−ωs We use the variables s and p to suggest “sum” and “product”. The Φ de- ω termine G in the following sense. Analysable instances of µ-synthesis 5 Proposition2.3. Foreveryω ∈T,Φ mapsGanalyticallyintoD.Conversely, ω if (s,p)∈C2 is such that |Φ (s,p)|<1 for all ω ∈T, then (s,p)∈G. ω Both statements can be derived from the identity |2−z−w|2−|2zw−z−w|2 =2(1−|z|2)|1−w|2+2(1−|w|2)|1−z|2. See [11, Theorem 2.1] for details. There is an analogous statement for Γ, but there are some subtleties. For one thing Φ is undefined at (2ω¯,ω¯2)∈Γ when ω ∈T. ω Proposition 2.4. For every ω ∈ T, Φ maps Γ\{(2ω¯,ω¯2)} analytically into ω ∆. Conversely, if (s,p)∈C2 is such that |Φ (rs,r2p)|<1 for all ω ∈T and ω 0<r <1 then (s,p)∈Γ. In the second statement of the proposition the parameter r is needed: itdoesnotsufficethat|Φ (s,p)|≤1forallω ∈T(inthe casethatp=1the ω last statement is true if and only if s ∈ R, whereas for (s,p) ∈ Γ, of course |s|≤2). We found the functions Φ by applying Agler’s theory of families of ω operator tuples [5, 6]. We studied the family F of commuting pairs of op- erators for which Γ is a spectral set, and its dual cone F (that is, the ⊥ collection of hereditary polynomials that are positive on F). Agler had pre- viously done the analogous analysis for the bidisc, and shown that the dual cone was generated by just two hereditary polynomials; this led to his cele- brated realization theorem for bounded analytic functions on the bidisc. On incorporating symmetry into the analysis we found that the cone F had ⊥ the 1-parameterfamily of generators1−Φ Φ , ω ∈T. From this fact many ∨ω ω conclusions follow: see [13] for more on these ideas. Operatortheoryplayedanessentialroleinourdiscoveryofthefunctions Φ .Oncetheyareknown,however,thegeometryofGandΓcanbedeveloped ω without the use of operator theory. 2.2. A necessary condition Suppose that F is a solution of the spectral Nevanlinna-Pick problem (2.1), (2.2) with k = 2. Let us write s = trW , p = detW for j = 1,...,n. For j j j j any ω ∈T and 0<t<1 the composition D−t→F Σo (t−r,→det)G−Φ→ω D is an analytic self-map of D under which 2ωt2p −ts λ 7→Φ (ts ,t2p )= j j for j =1,...,n. j ω j j 2−ωts j Thus, by Pick’s Theorem, n 1−Φ (ts ,t2p )Φ (ts ,t2p ) ω i i ω j j ≥0. (2.6) 1−λ¯ λ (cid:20) i j (cid:21)i,j=1 6 N. J. Young On conjugating this matrix inequality by diag{2−ωts } and letting α=tω j we obtain the following necessary condition for the solvability of a 2 × 2 spectral Nevanlinna-Pick condition [5, Theorem 5.2]. Theorem 2.5. If there exists an analytic map F : D → Σ satisfying the equations F(λ )=W for j =1,...,n j j and r(F(λ))≤1 for all λ∈D then, for every α such that |α|≤1, n (2−αs )(2−αs )−|α|2(2αp −s )(2αp −s ) i j i i j j ≥0 (2.7) " 1−λ¯iλj # i,j=1 where s =trW , p =detW for j =1,...,n. j j j j In the case that the W all have spectral radius strictly less than one, j thecondition(2.7)holdsforallα∈∆ifandonlyifitholdsforallα∈T,and hence the condition only needs to be checked for a one-parameter pencil of matrices. It is of course less simple than the classical Pick condition in that it comprisesaninfinite collectionofalgebraicinequalities,but itis neverthe- less checkable in practice with the aid of standard numerical packages. Its majordrawbackis that it is not sufficient for solvability of the 2×2 spectral Nevanlinna-Pick problem. Example 2.6. Let 0<r <1 and let λ2 λ(λ3+r) h(λ)= 2(1−r) , . 1+rλ3 1+rλ3 (cid:18) (cid:19) Let λ ,λ ,λ be any three distinct points in D and let h(λ ) = (s ,p ) for 1 2 3 j j j j =1,2,3.Wecanprove[3]that,inanyneighbourhoodof(s ,s ,s )in(2D)3, 1 2 3 there exists a point (s ,s ,s ) such that (s ,p ) ∈ G, the Nevanlinna-Pick ′1 ′2 ′3 ′j j data λ 7→Φ (s ,p ), j =1,2,3, j ω ′j j are solvable for all ω ∈T, but the Nevanlinna-Pick data λj 7→Φm(λj)(s′j,pj), j =1,2,3, are unsolvable for some Blaschke factor m. It follows that the interpolation data λ 7→(s ,p ), j =1,2,3, j ′j j satisfy the necessary condition of Theorem 2.5 for solvability, and yet there is no analytic function h:D→Γ such that h(λ )=(s ,p ) for j =1,2,3. j ′j j Hence, if we choose nonscalar 2 × 2 matrices W ,W ,W such that 1 2 3 (trW ,detW ) = (s ,p ), then the spectral Nevanlinna-Pick problem with j j j j data λ 7→ W satisfies the necessary condition of Theorem 2.5 and yet has j j no solution. See also [22] for another example. Analysable instances of µ-synthesis 7 2.3. Two points and two-by-two matrices When n=k =2the conditioninTheorem2.5is sufficientfor the solvability of the spectral Nevanlinna-Pick problem. We shall now prove the main theorem from Section 1. Recall the state- ment: Theorem1.1. Let λ ,λ ∈D be distinct points, let W ,W benonscalar 2×2 1 2 1 2 matrices of spectral radius less than 1 and let s = trW , p = detW for j j j j j =1,2. The following three statements are equivalent: (1) there exists an analytic function F :D→C2 2 such that × F(λ )=W , F(λ )=W 1 1 2 2 and r(F(λ))≤1 for all λ∈D; (2) (s p −s p )ω2+2(p −p )ω+s −s λ −λ 2 1 1 2 2 1 1 2 1 2 max ≤ ; (2.8) ω∈T (cid:12)(s1−s¯2p1)ω2−2(1−p1p¯2)ω+s¯2−s1p¯2(cid:12) (cid:12)1−λ¯2λ1(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (3) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 2 (2−ωs )(2−ωs )−(2ωp −s )(2ωp −s ) i j i i j j ≥0 (2.9) " 1−λ¯iλj # i,j=1 for all ω ∈T. The proof depends on some elementary notions from the theory of invariant distances.Agoodsourceforthe generaltheoryis[35],buthereweonlyneed the following rudiments. We denote by d the pseudohyperbolic distance on the unit disc D: λ −λ d(λ ,λ )= 1 2 for λ ,λ ∈D. 1 2 1−λ¯ λ 1 2 (cid:12) 2 1(cid:12) For any domain Ω∈Cn we de(cid:12)(cid:12)fine the Le(cid:12)(cid:12)mpert function δΩ :Ω×Ω→R+ by (cid:12) (cid:12) δ (z ,z )=infd(λ ,λ ) (2.10) Ω 1 2 1 2 over all λ ,λ ∈ D such that there exists an analytic map h : D → Ω such 1 2 that h(λ ) = z and h(λ ) = z . We define1 the Carath´eodory distance C : 1 1 2 2 Ω Ω×Ω→R+ by C (z ,z )=supd(f(z ),f(z )) (2.11) Ω 1 2 1 2 over all analytic maps f :Ω→D. If Ω is bounded then C is a metric on Ω. Ω It is not hard to see (by the Schwarz-Pick Lemma) that C ≤ δ for Ω Ω any domain Ω. The two quantities C , δ are not always equal – the punc- Ω Ω tured disc provides an example of inequality. The question of determining thedomainsΩforwhichC =δ isoneoftheconcernsofinvariantdistance Ω Ω theory. 1ConventionallythedefinitionoftheCarath´eodorydistancecontainsatanh−1ontheright handsideof (2.11).Forpresentpurposesitisconvenient toomitthetanh−1. 8 N. J. Young Proof. Let z =(s ,p )∈G. j j j (1)⇔(2) In view of Proposition 2.2 we must show that the inequality (2.8) is equivalent to the existence of an analytic h : D → Γ such that h(λ ) = z j j forj =1,2.By definitionofthe Lempertfunction δG,suchanh existsif and only if δG(z1,z2)≤d(z1,z2). By [11, Corollary5.7]we have δG =CG,and by [11, Theorem1.1 andCorol- lary 3.4], CG(z1,z2)=maxd(Φω(z1),Φω(z2)) (2.12) ω T ∈ (s p −s p )ω2+2(p −p )ω+s −s =max 2 1 1 2 2 1 1 2 . ω∈T (cid:12)(s1−s¯2p1)ω2−2(1−p1p¯2)ω+s¯2−s1p¯2(cid:12) (cid:12) (cid:12) Thus the desired function(cid:12) h exists if and only if the inequality (2.8)(cid:12)holds. (cid:12) (cid:12) (2)⇔(3) By equation (2.12), the inequality (2.8) is equivalent to d(Φ (z ),Φ (z ))≤d(λ ,λ ) for all ω ∈T. ω 1 ω 2 1 2 BytheSchwarz-PickLemma,thisinequalityholdsifandonlyif,forallω ∈T, there exists a function f in the Schur class such that f (λ ) = Φ (z ) for ω ω j ω j j =1,2. By Pick’s Theorem this in turn is equivalent to the relation 1−Φ¯ (z )Φ (z ) 2 ω i ω j ≥0. 1−λ¯ λ (cid:20) i j (cid:21)i,j=1 Conjugate by diag{2−ωs ,2−ωs } to obtain (2)⇔(3). (cid:3) 1 2 Remark 2.7. If one removes the hypothesis that W ,W be nonscalar from 1 2 Theorem 1.1 one can still give a solvability criterion. If both of the W j are scalar matrices then the problem reduces to a scalar Nevanlinna-Pick problem. If W = cI and W is nonscalar then the corresponding spectral 1 2 Nevanlinna-Pick problem is solvable if and only if r((W −cI)(I −c¯W ) 1)≤d(λ ,λ ) 2 2 − 1 2 (see [7, Theorem 2.4]). This inequality can also be expressed as a somewhat cumbersomealgebraicinequalityinc,s ,p andd(λ ,λ )[7,Theorem2.5(2)]. 2 2 1 2 2.4. Ill-conditioned problems The results of the preceding subsection suggest that solvability of spectral Nevanlinna-Pickproblems depends onthe derogatorystructure of the target matrices – that is, in the case of 2 × 2 matrices, on whether or not they are scalar matrices. It is indeed so, and in consequence problems in which a target matrix is close to scalar can be very ill-conditioned. Example 2.8. [7, Example 2.3] Let β ∈D\{0} and, for α∈C let 0α 0 β W (α)= , W = . 1 00 2 0 2β (cid:20) (cid:21) (cid:20) 1+β(cid:21) Analysable instances of µ-synthesis 9 Consider the spectral Nevanlinna-Pick problem with data 0 7→ W (α), β 7→ 1 W .Ifα=0thenthe problemis notsolvable.Ifα6=0,however,by Proposi- 2 tion2.2theproblemissolvableifandonlyifthereexistsananalyticfunction f :D→Γ such that 2β f(0)=(0,0) and f(β)= . 1+β It may be checked [8] that 2(1−β)λ λ(λ−β) f(λ)= , 1−βλ 1−βλ (cid:18) (cid:19) issuchafunction.ThustheproblemhasasolutionF foranyα6=0.Consider α a sequence (α ) of nonzero complex numbers tending to zero: the functions n F cannot be locally bounded, else they would have a cluster point, which αn would solve the problem for α = 0. If α is, say, 10 100 then any numeri- − cal method for the spectral Nevanlinna-Pick problem is liable to run into difficulty in this example. 2.5. Uniqueness and the construction of interpolating functions Problem SNP never has a unique solution. If F is a solution of Problem SNP then so is P 1FP for any analytic function P : D → Ck k such that − × P(λ) is nonsingular for every λ ∈ D and P(λ ) is a scalar matrix for each j interpolation point λ . There are always many such P that do not commute j with F, save in the trivialcase that F is scalar.Nevertheless,the solution of thecorrespondinginterpolationproblemforΓcanbe unique.Consideragain the case n = k = 2 with W ,W nonscalar. By Theorem 1.1, the problem 1 2 is solvable if and only if inequality (2.8) holds. In fact it is solvable uniquely if and only if inequality (2.8) holds with equality. This amounts to saying that each pair of distinct points of G lies on a unique complex geodesic of G, which is true by [12, Theorem 0.3]. (An analytic function h : D → G is a complex geodesic of G if h has an analytic left-inverse). Moreover, in this case the unique analytic function h : D → G such that h(λ ) = (s ,p ) for j j j j =1,2 can be calculated explicitly as follows [11, Theorem 5.6]. Chooseanω ∈Tsuchthatthe maximumonthelefthandsideof (2.8) 0 is attained at ω . Since equality holds in (2.8), we have 0 d(Φ (z ),Φ (z ))=d(λ ,λ ), ω0 1 ω0 2 1 2 where z = (s ,p ). Thus Φ is a Carath´eodory extremal function for the j j j ω0 pair ofpoints z ,z in G.It is easy (for example,by Schurreduction) to find 1 2 the unique Blaschke product p of degree at most 2 such that p(λ )=p , p(λ )=p and p(ω¯ )=ω¯2). 1 1 2 2 0 0 Define s by ω p(λ)−λ s(λ)=2 0 for λ∈D. 1−ω λ 0 def Then h = (s,p) is the required complex geodesic. 10 N. J. Young Note that h is a rational function of degree at most 2. It can also be expressedin the formofa realization:h(λ)=(trH(λ),detH(λ)) where H is a 2×2 function in the Schur class given by H(λ)=D+Cλ(1−Aλ)−1B AB fora suitableunitary 3×3or4×4matrix givenby explicitformulae CD (cid:20) (cid:21) (see [4], [12, Theorem 1.7]). 2.6. More points and bigger matrices Our hope in addressing the case n = k = 2 of the spectral Nevanlinna-Pick problem was of course that we could progress to the general case. Alas, we have not so far managed to do so. We have some hope of giving a good solvability criterion for the case k = 2, n = 3, but even the case n = 4 appears to be too complicated for our present methods. The case of two points and k × k matrices, for any k, looks at first sightmorepromising.Thereisanobviouswaytogeneralizethesymmetrized bidisc: we define the open symmetrized polydisc G to be the domain k G ={(σ (z),...,σ (z)):z ∈Dk}⊂Ck k 1 k where σ denotes the elementary symmetric polynomial in z = (z1,...,zk) m for 1≤m≤k. Similarly one defines the closed symmetrized polydisc Γ . As k in the case k =2, one can reduce Problem SNP to an interpolation problem for functions from D to Γ under mild hypotheses onthe targetmatrices W k j (specifically, that they be nonderogatory).However, the connection between ProblemSNP andthe correspondinginterpolationproblems for Γ are more k complicated for k > 2, because there are more possibilities for the rational canonical forms of the target matrices [37]. The analogues for Γ of the Φ k ω were described by D. J. Ogle [39] and subsequently other authors, e.g. [23, 29]. Ogle generalized to higher dimensions the operator-theoretic method of [6] and thereby obtained a necessary condition for solvability analogous to Theorem 2.5. The solvability ofProblemSNP whenn=2 is genericallyequivalentto the inequality δGk(z1,z2)≤d(λ1,λ2) where z is the ktuple of coefficients in the characteristic polynomial of W . j j All we need is an effective formula for δG . It turns out that this is a much k harder problem for k > 2. In particular, it is false that δG = CG when k k k >2. This discovery [38] was disappointing, but not altogether surprising. There is another type of solvability criterion for the 2 × 2 spectral Nevanlinna-Pick problem with general n [10, 14], but it involves a search over a nonconvex set, and so does not count for the purpose of this paper as an analytic solution of the problem. Another paper on the topic is [24]. It is heartening that the study of the complex geometry and analysis of the symmetrized polydisc has been taken up by a number of specialists inseveralcomplex variables,including G. Bharali,C.Costara,A. Edigarian,