The complex geomety of a domain related to $\mu$-synthesis

We describe the basic complex geometry and function theory of the {\em pentablock} $\mathcal{P}$, which is the bounded domain in $\mathbb{C}^3$ given by \[ \mathcal{P}= \{(a_{21}, \mathrm{tr} A, \det A): A= \begin{bmatrix} a_{ij}\end{bmatrix}_{i,j=1}^2 \in \mathbb{B}\} \] where $\mathbb{B}$ denotes the open unit ball in the space of $2\times 2$ complex matrices. We prove several characterizations of the domain. We describe its distinguished boundary and exhibit a $4$-parameter group of automorphisms of $\mathcal{P}$. We show that $\mathcal{P}$ is intimately connected with the problem of $\mu$-synthesis for a certain cost function $\mu$ on the space of $2\times 2$ matrices defined in connection with robust stabilization by control engineers. We demonstrate connections between the function theories of $\mathcal{P}$ and $\mathbb{B}$. We show that $\mathcal{P}$ is polynomially convex and starlike.


Introduction
In this paper we establish the basic complex geometry and function theory of the domain (1.1) P = {(a 21 , tr A, det A) : A = a ij 2 i,j=1 ∈ B} where B denotes the open unit ball in the space C 2×2 of 2×2 complex matrices, with the usual operator norm. We call this domain the pentablock. The name alludes to the fact that P ∩ R 3 is a convex body bounded by five faces, three of them flat and two curved (Theorem 9.3). P is a holomorphic image of the Cartan domain B. It is polynomially convex and starlike about the origin, but neither circled nor convex. The paper contains several characterizations of the domain, and descriptions of its distinguished boundary and of a 4-parameter group of automorphisms and of connections with the function theory of B.
The domain P arises in connection with the structured singular value, a cost function on matrices introduced by control engineers in the context of robust stabilization with respect to modelling uncertainty [13]. The structured singular value is denoted by µ, and engineers have proposed an interpolation problem called the µ-synthesis problem that arises from this source. Attempts to solve cases of this interpolation problem have led to the study of two other domains, the symmetrised bidisc [5] and the tetrablock [1], in C 2 and C 3 respectively, which have turned out to have many properties of interest to specialists in several complex variables [21,16,15] and to operator theorists [10,24]. The relationship between P and an instance of µ is explained in Section 5, and there is a more thoroughgoing discussion in the Conclusions (Section 13).
We shall denote the open unit disc by D, its closure by ∆ and the unit circle by T. The polynomial map implicit in the definition (1.1) will be written (1.2) π(A) = (a 21 , tr A, det A) where A = a ij 2 i,j=1 ∈ C 2×2 . Thus P = π(B). For the µ in question it transpires that µ(A) < 1 if and only if π(A) ∈ P. This statement is contained in Theorem 5.2, one of the main results of the paper. To illustrate the flavour of our results, here are foretastes of Theorem 5.2 and Theorem 7.1. Theorem 1.1. Let (s, p) = (λ 1 + λ 2 , λ 1 λ 2 ) where λ 1 , λ 2 ∈ D. Let a ∈ C and let The following statements are equivalent.
In this statement the cost function µ on C 2×2 is defined in Section 3, and Ψ z is the linear fractional map Ψ z (a, s, p) = a(1 − |z| 2 ) 1 − sz + pz 2 .
The significance of the equivalence of (1) and (2) is explained in the concluding section.

The symmetrised bidisc and the pentablock
The pentablock is closely related to the symmetrised bidisc, which is the domain in C 2 . Indeed, it is clear from the definition (1.1) that P is fibred over G by the map (a, s, p) → (s, p), since if A ∈ B then the eigenvalues of A lie in D and so (tr A, det A) ∈ G. Some basic properties of G will be needed, in particular the following characterizations [5].
Theorem 2.1. For a point (s, p) ∈ C 2 the following statements are equivalent.
If a ∈ C satisfies then (a, s, p) ∈ P.
In the case that |a| ≤ |w| choose ζ ∈ T such that λ 1 − λ 2 = ζ|λ 1 − λ 2 | and let Then π(A) = (a, s, p), and a simple calculation shows that We have shown that (a, s, p) ∈ P in the cases c − < |a| < c + and |a| ≤ |w|. The proposition will follow if we can show that This inequality is true. Let ρ denote the pseudohyperbolic distance from λ 1 to λ 2 : which is also true. Thus (a, s, p) ∈ P for all a such that |a| < 1 2 |1 −λ 2 λ 1 | + 1 2 Λ. The converse of Proposition 2.3 is also true (Theorem 5.2). Thus the fibre of P over the point (λ 1 + λ 2 , λ 1 λ 2 ) is the open disc of radius The closureP of P will also play a role; call it the closed pentablock. It is elementary thatP is the image of the closureB of B under π.
We denote by Γ the closure of G in C 2 , so that If a ∈ C satisfies then (a, s, p) ∈P.
3. An instance of µ and an associated domain The structured singular value µ E of A ∈ C m×n corresponding to subspace E of C n×m is defined by In the cases that 1) E comprises the whole of C n×m and 2) m = n and E consists of the scalar multiples of the identity, µ E is a familiar object, to wit the operator norm and the spectral radius respectively. When E comprises the diagonal matrices, µ E is an intermediate cost function µ diag . In these three cases the corresponding µ-synthesis problem leads to the analysis of the classical Nevanlinna-Pick interpolation problem, the symmetrised polydisc and (when m = n = 2) the tetrablock respectively. In this paper we are concerned with the case that m = n = 2 and where s = tr A and p = det A.
Proof. For X = z w 0 z , We have µ E (A) < 1 ⇔ inf{ X : X ∈ E and det(1 − AX) = 0} > 1. (3.4) Suppose that µ E (A) < 1. It follows from the last equivalence that if |w| ≤ 1 − |z| 2 then the contraction X = z w 0 z satisfies det(1 − AX) = 0, that is, In particular, on taking w = 0, we find that 1 − sz + pz 2 = 0 for all z ∈ ∆, which is to say that (s, p) ∈ G. Furthermore, the inequation (3.5) implies that In particular, |1 − sz + pz 2 | is strictly positive on T, and consequently the function tends to ∞ as |z| → 1 and hence attains its infimum over D at a point of D. Necessity in the statement (3.2) follows. Conversely, suppose that (s, p) ∈ G and In particular, on letting z = 0, we have We wish to show that µ E (A) < 1. Consider X ∈ E and suppose that det(1 − AX) = 0 and X ≤ 1. We can write Clearly |v| ≤ 1. If |v| = 1 then w = 0 and so contrary to the assumption that (s, p) ∈ G. Hence we have |v| < 1. Moreover and so contrary to the hypothesis (3.6). This contradiction shows that X ∈ E and det(1− AX) = 0 together imply that X > 1. A compactness argument shows that the infimum of X over X ∈ E such that det(1 − AX) = 0 is greater than 1, or in other words, µ E (A) < 1. The characterization (3.3) follows by scaling. Observe that µ E (rA) = rµ E (A) and so µ E (A) ≤ 1 if and only if µ E (rA) < 1 for all r ∈ (0, 1).
P µ is the domain in C 3 given by Corollary 3.2 asserts that A ∈ C 2×2 satisfies A ∈ B µ if and only if π(A) ∈ P µ . A major result of the paper is that P µ = P (Theorem 5.2).

A class of linear fractional functions
Proposition 3.1 introduces some linear fractional functions that will play an important role in the paper. Recall from Theorem 2.1 that the general point of G can be written in the form (β +βp, p) for some β, p ∈ D.  Moreover the supremum of 1−|z| 2 |1−sz+pz 2 | over z ∈ D is attained uniquely at the point Proof. Let us first deal with the case that s = 0. We have, in terms of w = 1/z 2 , Clearly |w + p| > |w| − 1 when |w| > 1, p ∈ D, and so the right hand side is at most 1. On letting w → ∞ we see that the supremum is exactly 1, attained uniquely at w = ∞. Thus equation (4.1) is true when s = 0, attained only at z = 0, in agreement with equation (4.2) since here β = 0. Now suppose that s = 0. The definition of κ can also be written Let with u, v real valued and let We have, at any point other than a zero of h, At critical points of g in {z : |z| > 1}, We may solve these equations to obtain and hence Thus the critical points of g are the points z, |z| > 1, such that or equivalently whence alsos |z| 2 − 2pz − 2z +s = 0. From these two equations we deduce that (−2s + 2sp)z + (−2sp + 2s)z = 0 In terms of β = (s −sp)/(1 − |p| 2 ) the last equation becomes βz =βz. Note that β = 0 since s = 0. We therefore have z = rβ for some r ∈ R. By virtue of equation (4.4), r must satisfy 0 = s|z| 2 − 2z − 2pz + s = (β +βp)r 2 |β| 2 − 2rβ − 2prβ + β +βp = (β +βp)(r 2 |β| 2 − 2r + 1).
Hence g(z) > 1 as claimed. Since g = 0 on T and g(z) → 1 as z → ∞, it follows that the unique critical point z = rβ of g in {z : |z| > 1} is a global maximum for g, and so the maximum κ(s, p) of g on {z : |z| > 1} is indeed given by the value (4.1), as required. Moreover, on rewriting the critical point given by equation (4.5) in terms of the original variable z ∈ D, we find that the maximum of 1−|z| 2 On combining Propositions 3.1 and 4.2 we obtain the following description.

The domains P and P µ
The purpose of this section is to show that P = P µ and to give criteria for membership of the domain. One inclusion is easy.
The next result provides characterizations of points in P and asserts that P = P µ .
and let a ∈ C. The following statements are equivalent.

Elementary geometry of the pentablock
In this section we give some basic geometric properties of the pentablock P and its closure.
A domain Ω is said to be polynomially convex provided that, for each compact subset K of Ω, the polynomial hull K of K is contained in Ω. Proof. Let us first show thatP is polynomially convex. Let x ∈ C 3 \P. We must find a polynomial f such that |f | ≤ 1 onP and |f (x)| > 1.
Since P r is polynomially convex, K ⊂ P r = P r ⊂ P, and so P is polynomially convex.
It follows that P is a domain of holomorphy (for example [19,Theorem 3.4.2]). However, Theorem 9.3 shows that P does not have a C 1 boundary, and consequently much of the theory of pseudoconvex domains does not apply to P.

Some automorphisms of P
By an automorphism of a domain Ω in C n we mean a holomorphic map f from Ω to Ω with holomorphic inverse. Every bijective holomorphic self-map of Ω is in fact an automorphism [19]. For In the event that α ∈ D the rational function B α is called a Blaschke factor. A Möbius function is a function of the form cB α for some α ∈ D and c ∈ T. The set of all Möbius functions is the automorphism group Aut D of D.
All automorphisms of the symmetrised bidisc G are induced by elements of Aut D [17]. That is, they are of the form for some υ ∈ Aut D. See also [7,Theorem 4.1] for another proof of this result.
Theorem 7.1. The maps f ωυ , for ω ∈ T and υ ∈ Aut D, constitute a group of automorphisms of P under composition. Each automorphism f ωυ extends analytically to a neighbourhood ofP. Moreover, for all ω 1 , ω 2 ∈ T, υ 1 , υ 2 ∈ Aut D, and, for all ω ∈ T, υ ∈ Aut D, One can use Theorem 5.2 and straightforward calculations to prove these statements. In this paper we will take a different approach. We show in Propositions 7.2 to Corollary 7.5 below that this group is the image under a homomorphism induced by π of a group of automorphisms of B. Moreover the explicit formula (7.11) shows that every rational function f ωυ extends holomorphically to a neighbourhood ofP.
We ask: is χ(F ) the full group of automorphisms of P?

The distinguished boundary of P
Let Ω be a domain in C n with closureΩ and let A(Ω) be the algebra of continuous scalar functions onΩ that are holomorphic on Ω. A boundary for Ω is a subset C of Ω such that every function in A(Ω) attains its maximum modulus on C. It follows from the theory of uniform algebras [11, Corollary 2.2.10] that (at least whenΩ is polynomially convex, as in the case of P) there is a smallest closed boundary of Ω, contained in all the closed boundaries of Ω and called the distinguished boundary of Ω (or the Shilov boundary of A(Ω)). In this section we shall determine the distinguished boundary of P; we denote it by bP.
Clearly, if there is a function g ∈ A(P) and a point u ∈P such that g(u) = 1 and |g(x)| < 1 for all x ∈P \ {u}, then u must belong to bP. Such a point u is called a peak point ofP and the function g a peaking function for u.
The set of 2 × 2 unitary matrices is denoted by U(2).
If w = 0, then a = 0, and one can take If w = 0, then | a w | ≤ 1. We can choose η ∈ T such that a w η ∈ R, and choose θ ∈ R such that sin(2θ) = a w η. Then (a, s, p) = π(U). Hence π(U(2)) = K 1 . We shall use the notation D(a; r) to mean the open disc centred at a ∈ C with radius r > 0. Proof. To show that K 1 is a closed boundary for A(P) consider any f ∈ A(P). Then f •π ∈ A(B), where B is the 2×2 matrix ball. Since U(2) is the distinguished boundary of B, there exists U ∈ U(2) such that f • π attains its maximum modulus at U. Hence f attains its maximum modulus at π(U). Therefore π(U(2)) is a closed boundary for A(P). By Proposition 8.2, π(U(2)) = K 1 .
Let us show that K 0 is a closed boundary for A(P). Consider f ∈ A(P). Since K 1 is a closed boundary for A(P), there exists (s, p) ∈ bΓ such that f attains its maximum modulus on the disc D(0; 1 − 1 4 |s| 2 ) × {(s, p)} ⊂ ∂P, say at the point (a, s, p). Then f must also attain its maximum modulus at a point (a 0 , s, p) for some a 0 such that |a 0 | = 1 − 1 4 |s| 2 . Otherwise |f (a, s, p)| > sup It follows that, for some r ∈ (0, 1) sufficiently close to 1, |f (ra, rs, rp)| > sup Since f is analytic in a neighbourhood of the disc rD(0; 1 − 1 4 |s| 2 ) × {(rs, rp)}, which is a subset of P by the starlike property of P, this contradicts the maximum principle applied to f (·, rs, rp).
Thus f attains its maximum modulus at a point of K 0 . Hence K 0 is a closed boundary for A(P). Theorem 8.4. For x ∈ C 3 , the following are equivalent.
It will be helpful if we first recall the shape of the real symmetrised bidisc.
Thus the plane Im s = Im p = 0 intersects G in the interior of the isosceles triangle with vertices at (0, −1) and (±2, 1).
The figure indicates the values of the parameter β, where s = β +βp, on the sides of the triangle. At the vertex (0, −1), one can take β to be any real number.
Although P is not convex, P ∩ R 3 is.
Theorem 9.2. The real pentablock P ∩ R 3 is convex.

A Schwarz Lemma for a general µ
The classical Schwarz Lemma gives a solvability criterion for a two-point interpolation problem in D. There is a simple analogue for two-point µ-synthesis; it is general in terms the cost functions µ E to which it applies, but very special in terms of the interpolation conditions. In this section we consider a general linear subspace E of C n×m and the corresponding µ E on C m×n , as in equation (3.1).
Definition 10.1. Ω µ E is the domain in C m×n given by We shall denote by N the Nevanlinna class of functions on the disc [23] and if F is a matricial function on D then we write F ∈ N to mean that each entry of F belongs to N. It then follows from Fatou's Theorem that if F ∈ N is an m × n-matrix-valued function then lim r→1− F (rλ) exists for almost all λ ∈ T.
In the next section we shall seek a Schwarz lemma for P. One might try to deduce such a result from Proposition 10.3 by lifting maps from Hol(D, P) to Hol(D, Ω µ E ). However, Section 12 shows that the lifting problem is delicate, and a Schwarz Lemma for P cannot easily be derived in this way. 11. What is the Schwarz Lemma for P?
On taking the supremum of the left hand side over z ∈ D and invoking Proposition 4.2 we obtain the inequality (11.2).

Analytic lifting
In the present context the µ-synthesis problem is an interpolation problem for analytic functions from D to B µ . If H : D → B µ is an analytic function satisfying interpolation conditions H(λ j ) = W j for given points λ 1 , . . . , λ n ∈ D and target points W 1 , . . . , W n ∈ B µ , then h def = π • H : D → P is an analytic function that satisfies (12.1) h(λ j ) = π(W j ) for j = 1, . . . , n.
The idea is that interpolation problems for Hol(D, P) should be easier than those for Hol(D, B µ ), as the bounded 3-dimensional domain P is likely to have a more tractable geometry than the unbounded 4-dimensional domain B µ . If we can find h ∈ Hol(D, P) satisfying the interpolation conditions (12.1), does it follow that we can lift h to a function H ∈ Hol(D, B µ ) that solves the original interpolation problem? (For the analogous questions in the cases of the symmetrised bidisc and the tetrablock, the answer is roughly yes, though with a few technicalities). We shall say that H ∈ Hol(D, C 2×2 ) is an analytic lifting of h ∈ Hol(D,P) if π • H = h. We say that H is a Schur lifting of h if π • H = h and H belongs to the matricial Schur class Of course, if H is an analytic lifting of h then H ∈ Hol(D,B µ ) (see Corollary 3.2).
The lifting problem for Hol(D, P) is delicate, as the following three examples show.
Example 12.1. Let h(λ) = (λ, 0, λ). This h ∈ Hol(D, P) lifts to H ∈ S 2×2 given by Here H(λ) does not belong to the open matrix ball B for any λ ∈ D. Our construction in Proposition 2.3 above gives the following non-analytic lifting of (λ, 0, λ) ∈ P to B: For suppose H has this property. We can write for some g, η in Hol D. Since det H = λ we must have for λ ∈ D. This is a contradiction, since the right hand side has a simple zero at 0, while the left hand side has a zero of multiplicity at least 2. These examples point to the following result. We shall call the variety (2) α is a zero of 1 4 s 2 − p of multiplicity n and (3) α is a zero of a of multiplicity greater than n.

Proof. A function (12.2)
H = 1 2 s − η g a 1 2 s + η is a lifting of h = (a, s, p) ∈ Hol(D, P) to Hol(D, C 2×2 ) if and only if η, g ∈ Hol D and det H = p, that is, (1) to (3). Then α is a zero of the right hand side of equation (12.3) of odd multiplicity n, whereas α is a zero of η 2 of even multiplicity. This is a contradiction, and so necessity holds in Proposition 12.3.
Conversely, suppose that there is no α ∈ D such that (1) to (3) hold. Suppose that {α j } are the zeros of 1 4 s 2 − p of odd multiplicity n j . The assumption implies that a does not vanish to order n j + 1 at α j . Choose g ∈ Hol D such that (1) 1 4 s 2 − p − ga does not vanish at any α j , There are functions h ∈ Hol(D,P) that have an analytic lifting but no Schur lifting.
Example 12.4. The function h(λ) = ( 1 2 , 0, λ) ∈ Hol(D,P) has an analytic lifting but no Schur lifting. More generally, let a ∈ ∆\{0} and let ϕ, ψ be inner functions. The function h = (aψ, 0, ϕ) ∈ Hol(D,P) has an analytic lifting provided there is no point α ∈ D that is a common zero of ϕ, ψ and has odd multiplicity n for ϕ and multiplicity greater than n for ψ. However h has a Schur lifting if and only if ϕ has an analytic square root and ψ divides ϕ in H ∞ .
Proof. The statement about the existence of an analytic lifting of h follows from Proposition 12.3.
The upshot of Proposition 12.3 and the three examples is that the µ-synthesis problem for µ E and the interpolation problem for Hol(D,P) are quite closely related, but that the rich function theory of Hol(D,B) may not be helpful for their solution.

Conclusions
The genesis of this paper was an attempt to find a new case of the notoriously difficult µ-synthesis problem that is amenable to analysis. The µ-synthesis problem arises in H ∞ control theory, for example, in the problem of designing a robustly stabilising controller for plants which are subject to structured uncertainty [13,14]. Here µ denotes a cost function on the space of m×n complex matrices; as in Section 3, it is given by (13.1) 1 µ E (A) = inf{ X : X ∈ E and det(1 − AX) = 0} where E is a linear space of matrices of appropriate size. Previous attempts to find analysable instances of µ-synthesis have led to the study of two domains in C 2 and C 3 , the symmetrised bidisc G of Section 2 and the tetrablock (see for example [1,26]). These domains have turned out to have interesting functiontheoretic [3,20,22], operator-theoretic [4,10,9,24] and geometric properties [12,5,16,17,27]. Could there be a class of 'µ-related domains' which have similarly rich theories, and which would throw light on the µ-synthesis problem? In this paper we study the next natural case of µ, which results from taking the space E in equation (13.1) to be the space of 2 × 2 matrices spanned by the identity matrix and a Jordan cell. This choice leads to the pentablock P. As we have shown, P is indeed amenable to analysis, though there remain some fundamental questions about P. We list some of them below.
The µ-synthesis problem is an interpolation problem for the space Hol(D, Ω) for certain domains Ω ⊂ C d . One is given distinct points λ 1 , . . . , λ N ∈ D and target points w 1 , . . . , w N ∈ Ω and the task is to determine whether there exists F ∈ Hol(D, Ω) such that F (λ j ) = w j for j = 1, . . . , N, and if so to find such an F (actually the interpolation conditions in [13,14] are of a more general form). In the case that N = 2 this problem is central to hyperbolic geometry in the sense of Kobayashi [18], so one could describe the problem as belonging to hyperhyperbolic geometry. In µ-synthesis the domain Ω has the form Ω µ = {A ∈ C m×n : µ(A) < 1}. This is typically an unbounded nonconvex and hitherto unstudied domain, and so the construction of holomorphic maps from D to Ω µ is a challenge. In the cases that µ is the spectral radius and µ diag there is an effective technique of dimensionreduction.
Let us say that the polynomial rank of a domain Ω ⊂ C d is the smallest positive integer r such that there exists a polynomial map π : C d → C r and a domain Ω ′ ⊂ C r such that z ∈ C d belongs to Ω if and only if π(z) ∈ Ω ′ . More succinctly, π must satisfy Ω = π −1 (π(Ω)). Clearly r ≤ d, since we may choose π to be the identity map on C d . In contrast, in all the special cases of µ mentioned in this paper it turns out that the polynomial rank of Ω µ is less than the dimension of the domain. In particular, Corollary 3.2 shows that the polynomial rank of Ω µ E is at most 3. The idea is that, when the polynomial rank of Ω is less than its dimension, the geometry of the lower-dimensional domain may be more accessible than that of Ω itself. A strategy for the construction of interpolating functions from D to Ω is to find a map h ∈ Hol(D, π(Ω)) which satisfies h(λ j ) = π(w j ) for each j, and then to attempt to lift h modulo π to an interpolating function in Hol(D, Ω).
When Ω = Ω µ for some µ the problem has a further helpful feature: since µ E is no greater than the operator norm, for any subspace E, it is always the case that Ω µ contains the open unit ball of the ambient space of matrices. In all three of the special cases of interest it turns out that the images of Ω µ and the unit ball B under the dimension-reducing map π coincide. Now the geometry and function theory of the Cartan domain B is rich and long established, and there are numerous ways of contructing maps in Hol(D, B); for example one may use the homogeneity of B to construct an interpolating function H by the standard process of Schur reduction. Then π • H is a holomorphic function from D to π(B) satisfying interpolation conditions, and one may then try to find an analytic lifting of π • H to an element of Hol(D, Ω µ ) that satisfies the given interpolation conditions. This strategy has had some successes, admittedly modest, for the two special cases of µ mentioned above.
In this new case of µ the strategy again looks promising. The dimension-reducing map π here takes A ∈ C 2×2 to (a 21 , tr A, det A), and Theorem 5.2 shows that π −1 (π(B)) = B µ . Here π(B) is the pentablock and we write B µ rather that Ω µ . The strategy outlined above is in principle feasible. However, Sections 11 and 12 shows that the final step, the lifting of maps from Hol(D, P) to Hol(D, B µ ) is more subtle than in previous cases.
We end with some natural questions.
Do the Carathéodory distance and Lempert functions coincide on the pentablock? See [15] for a positive solution of the corresponding question for the tetrablock.
Is the pentablock an analytic retract of either B or B µ ? Is the pentablock homogeneous? The corresponding questions for the tetrablock have negative answers [25].
What are the automorphisms of the pentablock? Are they all of the form described in Theorem 7.1?
What are the magic functions of the pentablock? See [7] for the definition of magic function and for their use in determining the automorphisms of a domain.