A rich structure related to the construction of analytic matrix functions

We analyse two special cases of $\mu$-synthesis problems which can be reduced to interpolation problems in the set of analytic functions from the disc into the symmetrised bidisc and into the tetrablock. For these inhomogeneous domains we study the structure of interconnections between the set of analytic functions from the disc into the given domain, the matricial Schur class, the Schur class of the bidisc, and the set of pairs of positive kernels on the bidisc subject to a boundedness condition. We use the theories of Hilbert function spaces and of reproducing kernels to establish these connections. We give a solvability criterion for the interpolation problem that arises from the $\mu$-synthesis problem related to the tetrablock.


Introduction
Engineering provides some hard challenges for classical analysis. In signal processing and, in particular, control theory, one often needs to construct analytic matrix-valued functions on the unit disc D or right half-plane subject to finitely many interpolation conditions and to some subtle boundedness requirements. The resulting problems are close in spirit to the classical Nevanlinna-Pick problem, but established operator-or functiontheoretic methods which succeed so elegantly for the classical problem do not seem to help for even minor variants. For example, this is so for the spectral Nevanlinna-Pick problem [13,22], which is to construct an analytic square-matrix-valued function F in D that satisfies a finite collection of interpolation conditions and the boundedness condition sup λ∈D r(F (λ)) ≤ 1 for all λ ∈ D.
This problem is a special case of the µ-synthesis problem of H ∞ control, which is recognised as a hard and important problem in the theory of robust control [16,17]. Even the special case of the spectral Nevanlinna-Pick problem for 2 × 2 matrices awaits a definitive analytic theory.
A major difficulty in µ-synthesis problems is to describe the analytic maps from D to a suitable domain X ⊂ C n or its closure X . In the classical theory X is a matrix ball, and the realisation formula presents the general analytic map from D to X in terms of a contractive operator on Hilbert space; this formula provides a powerful approach to a variety of interpolation problems. In the µ variants X can be unbounded, nonconvex, inhomogeneous and non-smooth, properties which present difficulties both for an operatortheoretic approach and for standard methods in several complex variables.
In this paper we exhibit, for certain naturally arising domains X , a rich structure of interconnections between four naturally arising objects of analysis in the context of 2 × 2 analytic matrix functions on D. This rich structure combines with the classical realisation formula and Hilbert space models in the sense of Agler to give an effective method of constructing functions in the space Hol(D, X ) of analytic maps from D to X , and thereby of obtaining solvability criteria for two cases of the µ-synthesis problem.
The rich structure is summarised in the following diagram, which we call the rich saltire 1 for the domain X .

S 2×2
Left S X SE ' ' P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P The objects are defined as follows: S 2×2 is the 2× 2 matricial Schur class of the disc, that is, the set of analytic 2× 2 matrix functions F on D such that F (λ) ≤ 1 for all λ ∈ D; S 2 is the Schur class of the bidisc D 2 , that is, Hol(D 2 , D), and for all z, λ, w, µ ∈ D, is positive semidefinite on D 2 and is of rank 1. The arrows in diagram (1.1) denote mappings and correspondences that will be described in Sections 4 to 7.
In this paper we consider the rich saltire for two domains X : the symmetrised bidisc and the tetrablock, defined below. Whereas S 2×2 and S 2 are classical objects that have been much studied, Hol (D, X ) and R have been introduced and studied within the last two decades in connection with special cases of the robust stabilisation problem. The maps in the upper northeast triangle of the rich saltire for a domain X do not depend on X .
The closed symmetrised bidisc is defined to be the set The tetrablock is the domain The closure of E is denoted byĒ. The symmetrised bidisc arises naturally in the study of the spectral Nevanlinna-Pick problem for 2 × 2 matrix functions. In a similar way, the tetrablock arises from another special case of the µ-synthesis problem for 2 × 2 matrix functions [22]. Define Diag def = z 0 0 w : z, w ∈ C and, for a 2 × 2-matrix A, µ Diag (A) = (inf{ X : X ∈ Diag, 1 − AX is singular}) −1 .
The µ Diag -synthesis problem: given points λ 1 , . . . , λ n ∈ D and target matrices W 1 , . . . , W n ∈ C 2×2 one seeks an analytic 2 × 2-matrix-valued function F such that F (λ j ) = W j for j = 1, . . . , n, and µ Diag (F (λ)) < 1, for all λ ∈ D. This problem is equivalent to the interpolation problem for Hol(D, E) studied in this paper; see [1,Theorem 9.2]. Here Hol(D, E) is the space of analytic maps from the unit disc D to E.
In the case of the symmetrised bidisc a number of components of the rich saltire for Γ were presented by Agler and two of the present authors in [3]. Aspects of the rich saltire for Γ were used in [3, Theorem 1.1] to prove a solvability criterion for the 2 × 2 spectral Nevanlinna-Pick interpolation problem. In this paper we give the final picture of the rich saltire for the symmetrised bidisc.
In the case of the tetrablock, with the aid of the rich saltire we obtain a solvability criterion for the µ Diag -synthesis problem. A strategy to obtain the solvability criterion is as follows. Reduce the problem to an interpolation problem in the set of analytic functions from the disc to the tetrablock, induce a duality between the set Hol(D, E) and S 2 , then use Hilbert space models for S 2 to obtain necessary and sufficient conditions for solvability.
The main result of this paper is the existence of the rich saltire, and the principal application thereof is the equivalence of (1) and (3) in the following assertion. Theorem 1.1. Let λ 1 , . . . , λ n be distinct points in D, let W 1 , . . . , W n be 2 × 2 complex matrices such that (W j ) 11 (W j ) 22 = det W j for each j, and let (x 1j , x 2j , x 3j ) = ((W j ) 11 , (W j ) 22 , det W j ) for each j. The following three conditions are equivalent.
(1) There exists an analytic 2 × 2 matrix function F in D such that

2)
and (2) There exists a rational function x : D → E such that This result is a part of Theorem 8.1, which we establish in Section 8, and [1, Theorem 9.2] (Theorem 3.1). The necessary and sufficient condition for the existence of a solution of the µ Diag -synthesis problem for 2 × 2 matrix functions with n > 2 interpolation points is given in terms of the existence of positive 3n-square matrices N, M satisfying a certain linear matrix inequality in the data, but with the constraint that N have rank 1. This kind of optimization problem can be addressed with the aid of numerical algorithms (for example, [14]), though we observe that, on account of the rank constraint, it is not a convex problem.
The paper is organized as follows. Sections 2 and 3 describe the basic properties of the symmetrized bidisc Γ and the tetrablock E respectively. They also present known results on the reduction of a 2 × 2 spectral Nevanlinna-Pick problem to an interpolation problem in the space Hol(D, Γ) of analytic functions from D to Γ, and on the reduction of a µ Diagsynthesis problem to an interpolation problem in the space Hol(D, E) of analytic functions from D to E. In Section 4 we construct maps between the sets S 2×2 and S 2 using the linear fractional transformation F F (λ) (z), λ, z ∈ D, for F ∈ S 2×2 . Relations between S 2×2 and the set of analytic kernels on D 2 are given in Section 5. Section 6 presents the rich saltire (6.1) for the symmetrised bidisc. The rich saltire for the tetrablock (7.1) is described in Section 7. Here we present a duality between the space Hol(D, E) and a subset of the Schur class S 2 of the bidisc. In Section 8 we use Hilbert space models for functions in S 2 to obtain necessary and sufficient conditions for solvability of the interpolation problem in the space Hol(D, E).
The closed unit disc in C will be denoted by ∆ and the unit circle by T. The complex conjugate transpose of a matrix A will be written A * . The symbol I will denote an identity operator or an identity matrix, according to context. The C * -algebra of 2 × 2 complex matrices will be denoted by M 2 (C).

The symmetrized bidisc G
The open and closed symmetrized bidiscs are the subsets The sets G and Γ are relevant to the 2×2 spectral Nevanlinna-Pick problem because, for a 2 × 2 matrix A, if r(·) denotes the spectral radius of a matrix, 3) Accordingly, if F is an analytic 2 × 2 matrix function on D satisfying r(F (λ)) ≤ 1 for all λ ∈ D then the function (tr F, det F ) belongs to the space Hol(D, Γ) of analytic functions from D to Γ. A converse statement also holds: every ϕ ∈ Hol(D, Γ) lifts to an analytic 2 × 2 matrix function F on D such that (tr F, det F ) = ϕ and consequently r(F (λ)) ≤ 1 for all λ ∈ D [8, Theorem 1.1]. The 2 × 2 spectral Nevanlinna-Pick problem can therefore be reduced to an interpolation problem in Hol(D, Γ). There is a slight complication in the case that any of the target matrices are scalar multiples of the identity matrix; for simplicity we shall exclude this case in the present paper.
The relation (2.3) scales in an obvious way: for ρ > 0, Theorem 2.1. Let λ 1 , . . . , λ n be distinct points in D and let W 1 , . . . , W n be 2×2 matrices, none of them a scalar multiple of the identity. The following two statements are equivalent.

5)
and h(D) is relatively compact in G.
Certain rational functions play a central role in the analysis of Γ.
In particular, Φ is defined and analytic on D × Γ (since |s| ≤ 2 when (s, p) ∈ Γ), Φ extends analytically to (∆ × Γ)\{(z, 2z,z 2 ) : z ∈ T}. See [7] for an account of how Φ arises from operator-theoretic considerations. The 1-parameter family Φ(ω, ·), ω ∈ T, comprises the set of magic functions of the domain G. The notion of magic functions of a domain is explained in [10], but for this paper all we shall need is the fact that Φ(D × Γ) ⊂ ∆ and a converse statement: if w ∈ C 2 and |Φ(z, w)| ≤ 1 for all z ∈ D then w ∈ Γ; see for example [9, Theorem 2.1] (the result is also contained in [6, Theorem 2.2] in a different notation).
A Γ-inner function is the analogue for Hol(D, Γ) of inner functions in the Schur class. A good understanding of rational Γ-inner functions is likely to play a part in any future solution of the finite interpolation problem for Hol(D, Γ), since such a problem has a solution if and only if it has a rational Γ-inner solution (for example, [15,Theorem 4.2]  where bΓ denotes the distinguished boundary of Γ. By Fatou's Theorem, the radial limit (2.7) exists for almost all λ ∈ T with respect to Lebesgue measure. The distinguished boundary bΓ of G (or Γ) is theŠilov boundary of the algebra of continuous functions on Γ that are analytic in G. It is the symmetrisation of the 2-torus: bΓ = {(z + w, zw) : |z| = |w| = 1}. The royal variety R = {(2z, z 2 ) : |z| < 1} plays an important role in the theory of Γ-inner functions.

The tetrablock E
The open and closed tetrablock are the subsets and The tetrablock was introduced in [1] and is related to the µ Diag -synthesis problem. The following theorem was proved in [1, Theorem 9.2]. The following functions play a central role in the analysis of the tetrablock [1].
Definition 3.2. The functions Ψ, Υ : C 4 → C are defined for (z, x 1 , x 2 , x 3 ) ∈ C 4 such that x 2 z = 1 and x 1 z = 1 respectively by In particular Ψ and Υ are defined and analytic everywhere except when x 2 z = 1 and x 1 z = 1 respectively. Note that, for x ∈ C 3 such that x 1 x 2 = x 3 , the functions Ψ(·, x) and Υ(·, x) are constant and equal to x 1 and x 2 respectively. In this paper we will use the function Ψ to define certain maps in the rich saltire of the tetrablock. By [1, Theorem 2.4], we have the following statement.
By [1, Theorem 2.9], E is polynomially convex, and so the distinguished boundary bE of E exists and is theSilov boundary of the algebra A(E) of continuous functions on E that are analytic on E. We have the following alternative descriptions of bE [1, Theorem 7.1].
(i) x ∈ bE; (ii) x ∈ E and |x 3 | = 1; (iii) x 1 = x 2 x 3 , |x 3 | = 1 and |x 2 | ≤ 1; (iv) either x 1 x 2 = x 3 and Ψ(·, x) is an automorphism of D or x 1 x 2 = x 3 and By [1,Corollary 7.2], bE is homeomorphic to D × T. By a peak point of E we mean a point p for which there is a function f ∈ A(E) such that f (p) = 1 and |f (x)| < 1 for all x ∈ E \ {p}. for almost all λ ∈ T.
By Fatou's Theorem, the radial limit (3.5) exists for almost all λ ∈ T with respect to Lebesgue measure. Note that, for an E-inner function ϕ = (ϕ 1 , ϕ 2 , ϕ 3 ) : D → E, ϕ 3 is an inner function on D in the classical sense.
A finite interpolation problem for Hol(D, E) has a solution if and only if it has a rational Γ-inner solution -see Theorem 8.1.

A realisation formula
In this section we construct maps between the sets S 2×2 and S 2 . For Hilbert spaces H, G, U and V , an operator P such that and an operator X : V → U for which I − P 22 X is invertible, we denote by F P (X) the linear fractional transformation F P (X) := P 11 + P 12 X(I − P 22 X) −1 P 21 F P (X) is an operator from H to G.
The following standard identity is a matter of verification. be operators from H ⊕ U to G ⊕ V . Let X and Y be operators from V to U for which I − P 22 X and I − Q 22 Y are invertible. Then Proposition 4.2. Let H, G, U and V be Hilbert spaces. Let P = P 11 P 12 P 21 P 22 be an operator from H ⊕ U to G ⊕ V and let X : V → U be an operator for which I − P 22 X is invertible. Then if X ≤ 1 and P ≤ 1 we have F P (X) ≤ 1.
Proof. By Proposition 4.1, Then By assumption, X ≤ 1 and P ≤ 1, and so I − X * X ≥ 0 and I − P * P ≥ 0.
Thus, for F = F ij 2 1 ∈ S 2×2 , the linear fractional transformation F F (λ) (z) is given by Proposition 4.6. The map SE is well defined.
Remark 4.7. In Definition 4.5, when either F 21 = 0 or F 12 = 0, the function is independent of z, and so in general the map SE can lose some information about F . However, in the case of the symmetrised bidisc, no information is lost; see Remark 6.15.

5.
Relations between S 2×2 and the set of analytic kernels on D 2 Basic notions and statements on analytic kernels can be found in the book [4] and in Aronszajn's paper [11].
Let N and M be analytic kernels on D 2 , and let K N,M be the hermitian symmetric function on D 2 × D 2 given by We define the set R 1 to be .
Proof. By definition, for z, λ, w, µ ∈ D. Clearly both N F and M F are analytic. To prove that (N F , M F ) ∈ R 1 one has to check that K N,M is an analytic kernel on D 2 of rank 1. Clearly K N,M is analytic. By Proposition 4.3, for all z, λ, w, µ ∈ D. Therefore for all z, λ, w, µ ∈ D. Thus K N F ,M F is an analytic kernel on D 2 of rank 1. Therefore Proof. For every F = F 11 F 12 0 F 22 ∈ S 2×2 , the functions γ and η are given by for all λ, z ∈ D. Thus, N F (z, λ, w, µ) = 0, for z, λ, w, µ ∈ D, and so has rank 0. Furthermore for z, λ, w, µ ∈ D, which is independent of z and w. Hence M F is a kernel on D 2 . Clearly both N F and M F are analytic. It is easy to see that for all z, λ, w, µ ∈ D, which is independent of z and w. Thus K N F ,M F is an analytic kernel on D 2 of rank 1. Therefore (N F , M F ) ∈ R 1 .
By Propositions 5.1 and 5.2, the map Upper E is well defined.

5.2.
Procedure U W and the set-valued map Upper W : R 11 → S 2×2 . Let F ∈ S 2×2 be such that F 21 = 0. Then the kernel N F has rank 1. In this case Upper E maps into a subset R 11 of R 1 rather than onto all of R 1 . By the Moore-Aronszajn Theorem [4, Theorem 2.23], for each kernel k on a set X, there exists a unique Hilbert function space H k on X that has k as its kernel.
Let us describe the procedure for the construction of a function in S 2×2 from a pair of kernels in R 11 .
for all z, λ, w, µ ∈ D and a function Ξ ∈ S 2×2 such that Hence (N, M ) ∈ R 11 can be presented in the following form and so for all z, λ, w, µ ∈ D. The left hand side of (5.4) can be written as , and the right hand side of (5.4) has the form Thus the relation (5.3) can be express by the statement that the Gramian of vectors  Hence there is an isometry Then, for z, λ ∈ D, we obtain the pair of equations Since L is a contraction, D ≤ 1 and for all z, λ ∈ D. Hence the first equation has the form Recall that, for the operator L, the linear fractional transformation Since L is a contraction, by Proposition 4.2 and Remark 4.4, and F L is analytic on D. Since A and Bλ(I H M − Dλ) −1 C are operators from C 2 to C 2 , F L is in S 2×2 . Then Ξ = F L has required properties.
The function Ξ constructed with Procedure U W is not necessarily unique since the functions f , g and v z,λ are not uniquely defined. The following proposition gives relations between different Ξ obtained using Procedure U W .
for all z, λ, w, µ ∈ D. Let Ξ 1 and Ξ 2 be constructed from (N, M ) using Procedure U W with the functions f 1 , g 1 , v 1 and f 2 , g 2 , v 2 , respectively. Then Proof. It is easy to see that f 2 = ζ f f 1 and g 2 = ζ g g 1 for some ζ f , ζ g ∈ T. By Theorem 5.5, Ξ 1 and Ξ 2 satisfy for all z, λ ∈ D. Hence and Since f 1 is a nonzero analytic function of 2 variables, the set of zeros of f 1 is nowhere dense in D 2 . Therefore Proposition 5.6 leads us to the following result.
Remark 5.16. The pair of kernels (N, M ) from Theorem 5.15 are known as Agler kernels for ϕ ∈ S 2 . There are papers with constructive proofs of the existence of Agler kernels. See for example [12], [20] and [21]. One can see that, for the Agler kernels (N, M ) for ϕ ∈ S 2 ,

Relations between Hol (D, Γ) and other objects in the rich saltire
The rich saltire for the symmetrized bidisc is the following.

S 2×2
Left S G SE ' ' P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P We will define maps of the rich saltire for G and describe connections between different maps in the diagram (6.1).
The following is trivial.  Definition 6.6. The map Lower E G : Hol (D, Γ) → S 2 is given by for h ∈ Hol (D, Γ).
Proposition 6.7. The map Lower E G is well defined.
One can ask the question: which subset of S 2 corresponds to Hol (D, Γ)? (6.2) If h = (s, p) ∈ Hol (D, Γ) then, for any fixed λ ∈ D, the map is a linear fractional self-map f (z) = az+b cz+d of D with the property "b = c". To make the last phrase precise, say that a linear fractional map f of the complex plane has the property "b = c" if f (0) = ∞ and either f is a constant map or, for some a, b and d in C, We shall denote the class of such functions f in S 2 by S b=c 2 .
The map Lower W G : S b=c 2 → Hol (D, Γ) is given by for all ϕ ∈ S b=c 2 . By Proposition 6.8, Lower W G is well defined.
and so for all z, λ ∈ D. Thus Lower E G • Lower W G (ϕ) = ϕ for all ϕ ∈ S b=c 2 . Therefore Lower W G is the inverse of Lower E G .
Let us consider how the defined maps interact with each other.
Proposition 6.11. The following holds SE • Left N G = Lower E G .
Proof. Let h ∈ Hol (D, Γ). Then, by Proposition 6.1, for Left N G (h) = F ∈ S 2×2 , Proof. By Proposition 6.11, SE • Left N G = Lower E G and, by Proposition 6.10, Lower W G is the inverse of Lower E G . The results follow immediately.
for all z, λ ∈ D and Left S G (F ) = (tr F, det F ) = (2F 11 , F 2 11 − F 21 F 12 ). Thus for all z, λ ∈ D. Therefore, for all F ∈ S 2×2 such that However for an arbitrary F ∈ S 2×2 we may have Lower E G • Left S G (F ) = SE (F ) as the following example shows. . Then F ∈ S 2×2 . It is easy to see that and Remark 6.15. In Definition 4.5, when either F 21 = 0 or F 12 = 0, the function is independent of z, and so in general the map SE can lose some information about F . However, in the case of the symmetrised bidisc, no information is lost. For h = (s, p) ∈ Hol (D, Γ) such that s 2 = 4p, by Definition 6.6, Secondly, by Definition 6.
6.4. The map SW G : R 11 → Hol (D, Γ). Proof. By Proposition 5.7, Therefore, for (N, M ) ∈ R 11 , where Ξ ∈ S 2×2 is a function constructed by Procedure UW for (N, M ). The later set is independent of the choice of Ξ.
Relations between SW G and other maps in the rich saltire are the following.
Proof. By Corollary 6.12, It is obvious that Left N G • Lower W G (ϕ) ∈ S 2×2 . By Proposition 5.13,

Relations between Hol (D, E) and other objects in the rich saltire
The rich saltire for the tetrablock is the following.

S 2×2
Left S E SE ' ' P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P We will define the maps of the rich saltire which depend on E and describe connections between the different maps in diagram (7.1). Proof. Consider first the case that x 1 x 2 = x 3 . By Proposition 3.3, |x 1 (λ)|, |x 2 (λ)| ≤ 1 for all λ ∈ D. Then the function

The map
is in S 2×2 and has the required properties (7.2) and (7.3), and moreover it is the only function with these properties. In the case that x 1 x 2 = x 3 , the H ∞ function x 1 x 2 − x 3 is nonzero, and so it has a unique inner-outer factorisation, say ϕe C = x 1 x 2 − x 3 where ϕ is inner, e C is outer and e C (0) ≥ 0. Let One can see that and |F 12 | = e Re 1 2 C = |F 21 | a. e. on T, F 21 is outer, and F 21 (0) ≥ 0. It follows that F is the only matrix satisfying the required properties (7.2) and (7.3).
Let us check that F ∈ S 2×2 . Clearly F is holomorphic on D. We must show that F (λ) ≤ 1 for all λ ∈ D. Let us prove that I − F (λ) * F (λ) is positive semidefinite for all λ ∈ D. It is enough to show that, for all λ ∈ D, the diagonal entries of I − F (λ) * F (λ) are non-negative and det (I − F (λ) * F (λ)) ≥ 0. Since |F 12 | = |F 21 | a. e. on T and a. e. on T. At almost every λ ∈ T, Let D 11 and D 22 be the diagonal entries of I − F * F . Since x(λ) ∈ E for λ ∈ D, by Proposition 3.3, for all λ ∈ D. Thus, for almost every λ ∈ T, By Proposition 3.3, for all λ ∈ D. Hence, for almost every λ ∈ T, for almost every λ ∈ T. Thus F (λ) ≤ 1 for almost every λ ∈ T, and so, by the Maximum Modulus Principle, F (λ) ≤ 1 for all λ ∈ D.
Definition 7.2. The map Left N E : Hol (D, E) → S 2×2 is given by (ii) Let us consider the following example: the function F on D which is defined by Clearly, F ∈ S 2×2 . Then and, by Definition 7.2, 7.3. The maps Lower E E : Hol (D, E) → S lf 2 and Lower W E : S lf 2 → Hol (D, E). Lemma 7.5. Let ϕ ∈ S 2 be such that ϕ(·, λ) is a linear fractional map for all λ ∈ D. Then ϕ can be written as , where a, b, c are functions from D to C, and b is analytic on D. Moreover, if c is analytic on D, then so is a.
Proof. Let ϕ ∈ S 2 be such that ϕ(·, λ) is a linear fractional map for all λ ∈ D. Then we can write where a, b, c, d are functions from D to C. Since ϕ ∈ S 2 , up to cancellation, ϕ(·, λ) does not have a pole at 0 for any λ ∈ D. Thus, without loss of generality, we may write ϕ(z, λ) = a(λ)z + b(λ) c(λ)z + 1 for all z, λ ∈ D. Moreover, since b(λ) = ϕ(0, λ) for all λ ∈ D, and so b is analytic on D.
Suppose c is analytic on D. Then for all z, λ ∈ D, and so a is analytic on D.
Definition 7.6. Let S lf 2 be the subset of S 2 which contains those ϕ for which ϕ(·, λ) is a linear fractional map of the form for all z, λ ∈ D, where c is analytic on D, and if a(λ) = b(λ)c(λ) for some λ ∈ D, then, in addition, |c(λ)| ≤ 1.
Proposition 7.7. Let ϕ be a function on D 2 . Then ϕ ∈ S lf 2 if and only if there exists a function x ∈ Hol (D, E) such that ϕ(z, λ) = Ψ(z, x(λ)) for all z, λ ∈ D.
By Proposition 7.7, the map below Lower E E is well defined.
One can use Proposition 7.7 to define the map Lower W E below.
(ii) for ϕ ∈ S lf 2 such that a = bc, and so ϕ(z, λ) = b(λ), z, λ ∈ D, Lower W E is the set map , where d is analytic and |d| ≤ 1 on D}.
Proposition 7.11. The following relations hold.

and so
Lower W E • Lower E E (x) = x.
Let us see how these maps interact with the other maps in the rich saltire (7.1).
Proposition 7.14. The equality Lower E E • Left S E = SE holds.
for all z, λ ∈ D. It follows that Lower E E • Left S E (F ) = SE(F ) for all F ∈ S 2×2 and so Lower E E • Left S E = SE as required.
The idea for SW E is that we want to follow Procedure UW with the application of the map Left S E to the function produced. The following proposition will facilitate this. Proof. By Proposition 5.7, a function F = for j = 1, . . . , n and k = 1, 2, 3. Write L as a block operator matrix where A, D act on C 2 , H respectively. Then, for j = 1, . . . , n and k = 1, 2, 3, we obtain the following equations From the second of these equations, for all j = 1, . . . , n and k = 1, 2, 3. Let Θ(λ) Since L is unitary and H is finite-dimensional, Θ is a rational 2 × 2 inner function. Hence the function x := (a, d, det Θ) is a rational E-inner function. We claim that x satisfies the interpolation conditions (8.2) x(λ j ) = (x 1j , x 2j , x 3j ) for all j = 1, . . . , n.
Theorem 8.1 gives us a criterion for the solvability of the interpolation problem find x ∈ Hol(D, E) such that x(λ j ) = (x 1j , x 2j , x 3j ) for j = 1, . . . , n. The process can be summarized as follows.
Procedure SW Let λ j and (x 1j , x 2j , x 3j ) be as in Theorem 8.1. Let z 1 , z 2 , z 3 be a triple of distint points in D, and N, M be positive 3n-square matrices such that rank (N ) ≤ 1 and the inequality (8.4) holds.
The criterion for the µ Diag -synthesis problem (Theorem 1.1) follows from Theorem 3.1 and Theorem 8.1. The tetrablock E is a bounded 3-dimensional domain, which is more amenable to study than the unbounded 4-dimensional domain Σ def = {A ∈ C 2×2 : µ Diag (A) < 1}.