aa r X i v : . [ m a t h . C V ] A ug THE AUTOMORPHISM GROUP OF THE TETRABLOCK
N. J. YOUNG
Abstract.
The tetrablock is shown to be inhomogeneous and its automorphism group isdetermined. A type of Schwarz lemma for the tetrablock is proved. The action of the au-tomorphism group is described in terms of a certain natural foliation by complex geodesicdiscs. Introduction
The tetrablock is the domain E in C defined by E = { ( x , x , x ) ∈ C : 1 − x z − x w + x zw = 0 for all z, w ∈ C such that | z | ≤ , | w | ≤ } . E is a non-convex domain whose intersection with R is a regular tetrahedron. It is of interestbecause of its relation to a certain function-theoretic problem that arises in control engineering;see Section 6 below. In this paper we answer three questions: is E homogeneous? Is E an analyticretract of the unit ball of the space of 2 × E ? Here an automorphism of a domain Ω is an analytic bijective self-map of Ω having ananalytic inverse.In Section 2 we prove a Schwarz lemma for E : we find necessary and sufficient conditions on y ∈ C for the existence of an analytic map ϕ : D → E , where D is the open unit disc, such that ϕ (0) = (0 , ,
0) and ϕ ′ (0) = y and we give a formula for a suitable ϕ . This result enables us toshow in Section 3 that E is inhomogeneous; the proof uses E. Cartan’s classification of boundedhomogeneous domains in C and a little elementary theory of J ∗ -algebras. We also show that E is not an analytic retract of any bounded symmetric homogeneous domain of dimension less than16. In Section 4 we determine the automorphism group of E , thereby verifying a conjecture madein [2]. In Section 5 we show that the action of the automorphism group of E can be understoodin terms of a certain natural foliation of E by analytic discs; the group permutes the leaves ofthis foliation transitively, and the orbits of the group are naturally parametrised by the interval[0 , E and the problem of “ µ -synthesis” from controlengineering is outlined in [2, Section 9] and references cited there.We shall denote the closure of E by ¯ E and the closed unit disc by ∆. We write O for the origin(0 , ,
0) in C . The automorphism group of a domain Ω will be denoted by Aut Ω. If H, K areHilbert spaces then L ( H, K ) denotes the linear space of bounded linear operators from H to K with the operator norm. C × denotes the space of 2 × C ∗ norm. An important role in the analysis of E is played by the map(1.1) π : C × → C : [ a ij ] ( a , a , det[ a ij ]) . We shall write S × for the set of analytic functions F : D → C × such that || F ( λ ) || < λ ∈ D . Math. Subject Classifications : 32M12 (primary), 30C80, 93D21 (secondary)
Let us recapitulate here some of the eleven different characterizations of E from [2]. For presentpurposes four will suffice. One of them uses the rational function(1.2) Ψ( z, x , x , x ) = x z − x x z − . Roughly speaking we identify x ∈ C with the linear fractional transformation Ψ( ., x ); then x ∈ E if and only if x corresponds to a linear fractional transformation whose supremum on D is less than one. This statement is not quite precise, since if x x = x then Ψ( ., x ) is constantand equal to x . Points x for which x x = x are called triangular points ; they require specialtreatment. Theorem 1.1.
For x = ( x , x , x ) ∈ C the following are equivalent. (1) x ∈ E ; (2) | x − ¯ x x | + | x x − x | < − | x | ; (3) sup z ∈ D | Ψ( z, x ) | < and if x x = x then, in addition, | x | < ; (4) there exists a symmetric matrix A ∈ C × such that || A || < and π ( A ) = x ; (5) there exist β , β ∈ C such that | β | + | β | < and x = β + ¯ β x , x = β + ¯ β x . For the proof see [2, Theorem 2.1]. Some basic complex geometry of E is described in thisreference. For example, E is starlike about the origin, is polynomially convex, has a distinguishedboundary of 3 real dimensions and admits a group of automorphisms of 6 real dimensions.Condition (iv) reveals a close connection between E and the two Cartan domains R I (2 ,
2) and R II (2), defined to be the open unit balls in the spaces of 2 × × E = π ( R I (2 , π ( R II (2)) . The homogeneity of R I (2 ,
2) was used in [2] to prove a Schwarz lemma for E , that is, a criterionfor the solvability of certain 2-point interpolation problems for analytic functions from D to E .In the next section a similar method is used to prove the other sort of Schwarz lemma for E : acriterion is found for the existence of an analytic function from D to E with a prescribed valueand derivative at a single point.2. A Schwarz lemma for the tetrablock
Theorem 2.1.
Let y ∈ C . There exists an analytic map ϕ : D → E such that ϕ (0) = O and ϕ ′ (0) = y if and only if (2.1) max {| y | , | y |} + | y | ≤ . Proof . Suppose such a ϕ exists. Write ϕ = ( ϕ , ϕ , ϕ ). By [2, Theorem 1.2], for any λ ∈ D ,max (cid:26) | ( ϕ − ¯ ϕ ϕ )( λ ) | + | ( ϕ ϕ − ϕ )( λ ) | − | ϕ ( λ ) | , | ( ϕ − ¯ ϕ ϕ )( λ ) | + | ( ϕ ϕ − ϕ )( λ ) | − | ϕ ( λ ) | (cid:27) ≤ | λ | . Divide through by | λ | and let λ → {| ϕ ′ (0) | + | ϕ ′ (0) | , | ϕ ′ (0) | + | ϕ ′ (0) |} ≤ . Since ϕ ′ (0) = y we have max {| y | , | y |} + | y | ≤ , and the inequality (2.1) is necessary for the existence of ϕ .Conversely, suppose that (2.1) holds. We can suppose that | y | ≥ | y | . If y = 0 then also y = 0, | y | ≤ ϕ ( λ ) = (0 , , λy ) is sufficient. We may therefore assume that y = 0. We shall construct F ∈ S × such that ϕ = π ◦ F has the desired properties. Note that HE AUTOMORPHISM GROUP OF THE TETRABLOCK 3 since π ( R I (2 , E , for F ∈ S × , ϕ maps D into E . Let ζ ∈ D be a number to be chosen laterand let F = [ F ij ] satisfy(2.2) F (0) = (cid:20) ζ (cid:21) . Then ϕ (0) = O and ϕ ′ (0) = ( π ◦ F ) ′ (0) = ( F ′ , F ′ , F ′ F + F F ′ − F ′ F − F F ′ )(0)= ( F ′ , F ′ , − ζF ′ )(0) . Thus ϕ ′ (0) = y if and only if(2.3) F ′ (0) = (cid:20) y ∗− y /ζ y (cid:21) . Accordingly our task is to find ζ ∈ D and F ∈ S × such that the equations (2.2) and (2.3) aresatisfied.We shall use the matricial M¨obius transformation M Z of 2 × × Z by M Z ( X ) = − Z + D Z ∗ X (1 − Z ∗ X ) − D Z , where D Z = (1 − Z ∗ Z ) . The transformation M Z is an automorphism of the unit ball R I (2 , C × , has inverse M − Z and maps Z to 0. We have, for any F ∈ S × ,( M Z ◦ F ) ′ = D Z ∗ [ F ′ (1 − Z ∗ F ) − + F (1 − Z ∗ F ) − Z ∗ F ′ (1 − Z ∗ F ) − ] D Z = D Z ∗ (1 − F Z ∗ ) − F ′ (1 − Z ∗ F ) − D Z . (2.4)Let(2.5) Z = (cid:20) ζ (cid:21) for some ζ ∈ D . Then Z is a strict contraction and D Z = " − | ζ | ) , D Z ∗ = " (1 − | ζ | )
00 1 . Hence, if F satisfies equations (2.2) and (2.3), then( M Z ◦ F ) ′ (0) = D − Z ∗ F ′ (0) D − Z = y (1 − | ζ | ) F ′ (0)1 − | ζ | − y ζ y (1 − | ζ | ) . (2.6)If the required F exists then, by the Schwarz Lemma for R I (2 , F exists by working back from equation (2.6).The choice ζ = p − | y | in equation (2.6) leads us to define(2.7) Y ( ξ ) = y | y | ξ − y p − | y | y | y | N. J. YOUNG for some ξ ∈ C . Since, by hypothesis, | y | ≤ − | y | , the first column of Y ( ξ ) has norm (cid:26) | y | + | y | − | y | (cid:27) ≤ (cid:26) | y | + | y | (1 − | y | )1 − | y | (cid:27) = {| y | + | y |} ≤ , and since | y | ≤ | y | , the second row of Y ( ξ ) also has norm at most 1. By Parrott’s Theorem([13] or [14, Theorem 12.22]) there exists ξ ∈ C such that || Y ( ξ ) || <
1; in fact, a suitable choiceis(2.8) ξ = y y ¯ y p − | y || y | (1 − | y | − | y | ) . Let H ( λ ) = λY ( ξ ) , λ ∈ D . Then H ∈ S × and H (0) = 0 , H ′ (0) = Y ( ξ ) . Define F = M − Z ◦ H , where as before Z is the right hand side of equation (2.2), and now ζ = p − | y | . Then F ∈ S × , F (0) = M − Z (0) = Z = (cid:20) ζ (cid:21) and F ′ (0) = ( M − Z ◦ H ) ′ (0) = ( D Z ∗ (1 + HZ ∗ ) − H ′ (1 + Z ∗ H ) − D Z )(0)= D Z ∗ Y ( ξ ) D Z = " | y |
00 1 y | y | ξ − y ζ y | y | " | y | = (cid:20) y ξ | y |− y /ζ y (cid:21) . On comparison with equations (2.2) and (2.3) we find that ϕ = π ◦ F satisfies the requirementsof the theorem.We can extract from the above proof an explicit formula for ϕ satisfying the conditions of thetheorem. Theorem 2.2.
Let y ∈ C be such that max {| y | , | y |} + | y | ≤ . Let ϕ : D → C be given by (2.9) ϕ ( λ ) = λ λ ¯ y C ( y ) ( y , y , C ( y ) λ + y ) where (2.10) C ( y ) = if y = y = 0 y y (1 − | y | ) | y | (1 − | y | − | y | ) if | y | ≤ | y | 6 = 0 y y (1 − | y | ) | y | (1 − | y | − | y | ) if | y | ≤ | y | 6 = 0 . Then ϕ is an analytic map from D to E , ϕ (0) = O and ϕ ′ (0) = y . Proof . We considered the case y = y = 0 in the proof of Theorem 2.1. Suppose without lossthat | y | ≤ | y | 6 = 0. It is immediate that ϕ as defined satisfies ϕ (0) = 0 , ϕ ′ (0) = y ; the task is toshow that ϕ is analytic and ϕ ( D ) ⊂ E . HE AUTOMORPHISM GROUP OF THE TETRABLOCK 5
Choose ζ = p − | y | and Z, ξ, Y = Y ( ξ ) as in equations (2.5), (2.8),(2.7) respectively, andlet F ( λ ) = M − Z ( λY ) = Z + D Z ∗ λY (1 + λZ ∗ Y ) − D Z = Z + λ ( D Z ∗ Y D Z )(1 + λZ ∗ D − Z ∗ Y D Z ) − . As we observed in the proof of Theorem 2.1, F ∈ S × . We have D Z ∗ Y D Z = (cid:20) y ξ | y |− y /ζ y (cid:21) , D − Z ∗ Y D Z = (cid:20) y | y | ξ − y /ζ y (cid:21) . Hence F ( λ ) = Z + λ ( D Z ∗ Y D Z ) (cid:18) λ (cid:20) ζ (cid:21) (cid:20) y | y | ξ − y /ζ y (cid:21)(cid:19) − = Z + λ λξζ (cid:20) y ξ | y |− y /ζ y (cid:21) (cid:20) λξζ − λζy | y | (cid:21) = (cid:20) ζ (cid:21) + λ λξζ (cid:20) y ξ | y | w y (cid:21) (2.11)where w = w ( λ ) = − y ζ (1 + λξζ ) − λζy y | y | = − y ζ − λζ y y | y | (1 − | y | − | y | )= − y p − | y | − λ p − | y | C ( y ) . (2.12)We find that det F ( λ ) = λ ( y y − wξ | y | )(1 + λξζ ) − λw p − | y | λξζ . Note that ξζ = y y ¯ y (1 − | y | ) | y | (1 − | y | − | y | ) = ¯ y C ( y ) . We have y y − w ( λ ) ξ | y | = y y (1 − | y | )1 − | y | − | y | + λ ¯ y | y | (cid:18) y y (1 − | y | )1 − | y | − | y | (cid:19) = | y | C ( y )(1 + λ ¯ y C ( y )) , and sodet F ( λ ) = λ λ ¯ y C ( y ) ( y + λ (1 − | y | ) C ( y )) + (cid:18) λ λ ¯ y C ( y ) (cid:19) | y | C ( y )(1 + λ ¯ y C ( y ))= λ λ ¯ y C ( y ) ( C ( y ) λ + y ) . Since F ∈ S × the map π ◦ F is analytic, maps D to E and satisfies (compare equation (2.11))( π ◦ F ) ( λ ) = F ( λ ) = λy λ ¯ y C ( y ) , ( π ◦ F ) ( λ ) = F ( λ ) = λy λ ¯ y C ( y ) , ( π ◦ F ) ( λ ) = det F ( λ ) = λ λ ¯ y C ( y ) ( C ( y ) λ + y ) . Comparison with equation (2.9) shows that π ◦ F = ϕ , and hence ϕ has the required properties. N. J. YOUNG
Remark 2.1.
In the event that the necessary condition of Theorem 2.1 holds with equality, thatis, max {| y | , | y |} + | y | = 1 , the function ϕ of Theorem 2.2 is a complex geodesic of E (that is, it has an analytic left inverse).Suppose that | y | ≤ | y | 6 = 0 and | y | + | y | = 1. Choose ω , ω ∈ T such that ω y = | y | , ω y = | y | ; then ω y + ω y = 1. For any z ∈ ∆ the rational function Ψ( z, . ) given by equation (1.2)maps E analytically into D . We haveΨ( ω, ϕ ( λ )) = ϕ ( λ ) ω − ϕ ( λ ) ϕ ( λ ) ω − λ ( Cλ + y ) ω − y λy ω − (1 + C ¯ y λ ) , where (since 1 − | y | = | y | ) C = C ( y ) = y y | y || y | ( | y | − | y | ) = y y | y | = y ¯ y . Choose ω = − ¯ ω ω . A little calculation gives the relationΨ( ω, ϕ ( λ )) = ¯ ω λ. Hence ω Ψ( ω, . ) : E → D is an analytic left inverse of ϕ , and so ϕ is a complex geodesic of E .One might expect (by analogy with the case of the unit disc) that in the extremal case ϕ should be E -inner [1], that is, the radial limit function of ϕ should map T almost everywhereinto the distinguished boundary b E of E . In fact b E is the intersection of the closure ¯ E of E withthe set { x ∈ C : | x | = 1 } [2, Theorem 7.1], and so an analytic map ϕ : D → E is E -inner if andonly if ϕ is a scalar inner function. For the function ϕ of the theorem, ϕ is inner if and only if y is “doubly extremal”, that is, | y | = | y | = 1 − | y | .3. The tetrablock is inhomogeneous
We shall show that the inhomogeneity of E follows from Theorem 2.1 and E. Cartan’s clas-sification of bounded homogeneous domains [7, page 313]. We use L. A. Harris’ theory of J ∗ -algebras[8]. A J ∗ -algebra is a closed subspace A of the Banach space L ( H, K ), for some Hilbertspaces
H, K , with the property that T ∈ A implies T T ∗ T ∈ A . The importance of such algebrashere is that, in dimensions up to 15, every bounded symmetric homogeneous domain is isomor-phic to the open unit ball of a J ∗ -algebra. A domain Ω is said to be symmetric if, for every z ∈ Ω, there is an analytic involution of Ω of which z is an isolated fixed point.A domain Ω is said to be an analytic retract of a domain Ω if there exist analytic maps h : Ω → Ω , κ : Ω → Ω such that κ ◦ h = id E . We define the indicatrix I (Ω , a ) of a domain Ωat a point a ∈ Ω to be the set I (Ω , a ) = { ϕ ′ (0) : ϕ is an analytic map from D to Ω , ϕ (0) = a } . It follows from the chain rule that if h : Ω ⊂ C n → Ω is analytic and a ∈ Ω then h ′ ( a ) I (Ω , a ) ⊂ I (Ω , h ( a )). If, further, h has an analytic left inverse κ , then κ ′ ◦ h ( a ) h ′ ( a ) is the identity operatoron C n , and so κ ′ ◦ h ( a ) is a linear operator that maps I (Ω , h ( a )) surjectively onto I (Ω , a ).We recall [9] that the rank of a J ∗ -algebra A is the supremum of the number of non-zeroelements in the spectrum of T ∗ T over all T ∈ A ; it is also equal to the maximum cardinality ofany set of mutually orthogonal non-zero minimal partial isometries in A [9, Corollary 5]. Everyfinite-dimensional J ∗ -algebra clearly has finite rank. Theorem 3.1. E is not an analytic retract of the open unit ball of any J ∗ -algebra of finite rank. HE AUTOMORPHISM GROUP OF THE TETRABLOCK 7
Proof . Let
A ⊂ L ( H, K ) be a J ∗ -algebra of rank r < ∞ , B its open unit ball. Suppose that h : E → B, κ : B → E are analytic and κ ◦ h = id E . Since the open unit ball of a J ∗ -algebrais homogeneous, we may replace h, κ by their compositions with automorphisms of B to ensurethat h (0) = 0. By Theorem 2.1, I ( E ,
0) = { y ∈ C : max {| y | , | y |} + | y | ≤ } . It is easy to see that I ( B,
0) is the closed unit ball ¯ B of A . Indeed, if T ∈ ¯ B then the function ϕ ( λ ) = λT maps D to B and 0 to 0 and satisfies ϕ ′ (0) = T , so that I ( B, ⊃ ¯ B , while if ϕ : D → B is analytic and maps 0 to 0 then the Schwarz lemma for D applied to the scalarfunctions h ϕ ( . ) ξ, η i , ξ ∈ H, η ∈ K shows that ϕ ′ (0) ∈ ¯ B . I ( E ,
0) is the closed unit ball of C with respect to the norm || y || E = max {| y | , | y |} + | y | . Since the linear operators h ′ (0) : C → A , κ ′ (0) : A → C are contractions with respect to || . || E , || . || A and κ ′ (0) h ′ (0) is the identity operator on C , it follows that h ′ (0) is an isometry. Let h ′ (0)(1 , ,
0) =
A, h ′ (0)(0 , ,
1) = B . Then for any λ, µ ∈ C ,(3.1) || λA + µB || L ( H,K ) = || h ′ (0)( λ, , µ ) || = || ( λ, , µ ) || E = | λ | + | µ | ;in particular, || A || = || B || = 1. By [9, Proposition 4], every element T ∈ A has a singular valuedecomposition T = m X k =1 s k V k where m ≤ r , each s k > V k ∈ A are mutually orthogonal non-zero minimal partialisometries. Let ω , ω , . . . be an infinite sequence of distinct points in T . Since || A + ω B || = 2we can write down the singular value decomposition A + ω B = 2 V + · · · + 2 V n + R where 1 ≤ n ≤ r, R ∈ A , || R || < R is orthogonal to V , . . . , V n . The space ofmaximising vectors of A + ω B is M = span { V ∗ K, . . . , V ∗ n K } . For x ∈ M we have 2 || x || = || Ax + ω Bx || ≤ || Ax || + || Bx || ≤ || x || . It follows that || Ax || = || x || = || Bx || . Moreover, the parallelogram law shows that Ax = ω Bx .Hence, for x ∈ M , 2 Ax = ( A + ω B ) x = 2 V x + · · · + 2 V n x. Thus we can write A = V + · · · + V n + A (3.2) ω B = V + · · · + V n + ω B (3.3)where A , B ∈ A , A and B are both orthogonal to V , . . . , V n and || A + ω B || <
2. For any ω ∈ T , ω = ω , || A + ωB || = 2 and A + ωB = (1 + ω ¯ ω )( V + · · · + V n ) + A + ωB . Since ω = ω we have | ω ¯ ω | <
2. Hence || A + ωB || = 2 for any ω ∈ T \ { ω } and by thesame arguments we have A = V n +1 + · · · + V n + A ,ω B = V n +1 + · · · + V n + ω B (3.4) N. J. YOUNG where A , B ∈ A , A and B are both orthogonal to V , . . . , V n and || A + ω j B || < , j = 1 , r steps. If we write W = V + · · · + V n etc. then, forsome N ≤ r , A = W + W + · · · + W N ,B = ¯ ω W + ¯ ω W + · · · + ¯ ω N W N where W , . . . , W N are mutually orthogonal non-zero partial isometries in A . Choose ω ∈ T different from ω , . . . , ω N : then || A + ωB || = || (1 + ω ¯ ω ) W + · · · + (1 + ω ¯ ω N ) W N || = max ≤ j ≤ N | ω ¯ ω j | < , contrary to equation (3.1). Hence the postulated maps h, κ do not exist. Corollary 3.2. E is inhomogeneous. Proof . E. Cartan showed that every bounded homogeneous domain in C is symmetric [7, page313]. Every bounded symmetric homogeneous domain in C n , n <
15, is the open unit ball of a J ∗ -algebra [8, Theorem 7]. E is bounded, and so if E is homogeneous then E is isomorphic to the openunit ball of a 3-dimensional J ∗ -algebra, contrary to Theorem 3.1. Hence E is inhomogeneous.4. The automorphism group of the tetrablock
Although the automorphism group Aut E does not act transitively on E , it is neverthelessquite large: there are commuting left and right actions of Aut D on E [2, Theorem 6.8]. Thesetwo actions together with the “flip” automorphism F : ( x , x , x ) ( x , x , x ) give a group G of automorphisms of E , and we conjectured in [2] that in fact G = Aut E . In this section weprove that the conjecture is correct.Roughly speaking, the actions of Aut D on E are by composition. Consider x ∈ ¯ E and y ∈ E .The linear fractional maps Ψ( ., x ) , Ψ( ., y ) given by equation (1.2) map ∆ into ∆ , D respectively[2, Theorems 2.4 and 2.7], and a simple calculation yields the relationΨ( ., x ) ◦ Ψ( ., y ) = Ψ( ., x ⋄ y )where x ⋄ y = 11 − x y ( x − x y , y − x y , x y − x y )(4.1) = (cid:18) Ψ( y , x ) , Ψ( x , F ( y )) , x y − x y − x y (cid:19) . We define x ⋄ y by equation (4.1) for any x, y ∈ C such that x y = 1. Consider υ ∈ Aut D :we can write υ = Ψ( ., τ ( υ )) for some τ ( υ ) ∈ ¯ E . The left action of Aut D on E or ¯ E is given by υ · x = τ ( υ ) ⋄ x , or equivalently Ψ( ., υ · x ) = υ ◦ Ψ( ., x ). Similarly one defines a right action by x · υ = x ⋄ τ ( υ ) , x ∈ E , υ ∈ Aut D . Fuller details of the construction are given in [2, Section 6]. If L υ , R χ for any υ, χ ∈ Aut D are given by L υ ( x ) = υ · x, R χ ( x ) = x · χ then L υ , R χ are commutingelements of Aut E . Moreover, L υ L χ = L υ ◦ χ and there is an involution υ υ ∗ on Aut D suchthat F L υ = R υ ∗ for any υ ∈ Aut D . It follows that(4.2) G def = { L υ R χ F ν : υ, χ ∈ Aut D , ν = 0 or 1 } is a subgroup of Aut E . Theorem 4.1.
Aut E = G. HE AUTOMORPHISM GROUP OF THE TETRABLOCK 9
The proof is based on the ideas of M. Jarnicki and P. Pflug in their determination of theautomorphism group of the symmetrised bidisc [10]; the author and J. Agler had previouslyfound a more elementary but longer proof of the same result. An important role in the proofis played by the rotations ρ ω ∈ Aut D , defined by ρ ω ( z ) = ωz. It is easy to show that, for any ω ∈ T and x ∈ E , ρ ω · x = ( ωx , x , ωx ) , x · ρ ω = ( x , ωx , ωx ) . Lemma 4.2.
Any automorphism of E that fixes every triangular point of E is the identityautomorphism of E . Proof . Let h ∈ Aut E fix all triangular points: h ( x , x , x x ) = ( x , x , x x ) for all x , x ∈ D .We have(4.3) h ′ ( O ) = a b c for some a, b, c ∈ C . For ω ∈ T let H ω be the element h − ◦ L ρ /ω ◦ h ◦ L ρ ω of Aut E . Then H ω ( O ) = ( O ) and H ′ ω ( O ) = h ′ ( O ) − diag(¯ ω, , ¯ ω ) h ′ ( O )diag( ω, , ω ) = b ( ω − . If H nω denotes the n th iterate of H ω then( H nω ) ′ ( O ) = H ′ ω ( O ) n = nb ( ω − . Now the isotropy group K of the origin in E , that is the group { f ∈ Aut E : f ( O ) = O } , iscompact with respect to the topology of locally uniform convergence, the map f f ′ ( O ) iscontinuous on Aut E and each H nω ∈ K . Hence the matrices ( H nω ) ′ ( O ) , n ≥ , are uniformlybounded. It follows that b = 0. Similarly we have a = 0 (replace L ρ ω by R ρ ω in the aboveargument). Thus h ′ ( O ) is diagonal and H ′ ω ( O ) is the identity matrix. By Cartan’s theorem (e.g.[12, Proposition 10.1.1]) H ω is the identity automorphism, and so h ◦ L ρ ω = L ρ ω ◦ h . Similarly h ◦ R ρ ω = R ρ ω ◦ h . If h = ( h , h , h ) then, for x ∈ E and ω ∈ T , h ( ωx , x , ωx ) = ( ωh ( x ) , h ( x ) , ωh ( x )) ,h ( x , ωx , ωx ) = ( h ( x ) , ωh ( x ) , ωh ( x )) . By the former equation, for fixed x , h is homogeneous of degree 0 in x , x while h , h arehomogeneous of degree 1 in x , x . Similarly, for fixed x , h is homogeneous of degree 0 and h , h are homogeneous of degree 1 in x , x . It follows that h ( x ) = ( αx , βx , γx x + δx )for some α, β, γ, δ ∈ C . Comparison with equation (4.3) shows that α = β = 1 , δ = c = 0. Since h fixes triangular points, γ = 1 − c and so h ( x ) = ( x , x , (1 − c ) x x + cx ). We must prove that c = 1.Observe that h and h − are polynomial maps and therefore extend continuously to ¯ E . Hence h induces an automorphism of the algebra A ( E ) of continuous scalar functions on ¯ E that areanalytic on E , and consequently h maps the Shilov boundary b E of A ( E ) to itself. According to [2,Theorem 7.1], x ∈ b E if and only if x = ¯ x x , | x | ≤ | x | = 1. For x = (¯ x x , x , x ) ∈ b E we have 1 = | h ( x ) | = | (1 − c )¯ x x x + cx | = (cid:12)(cid:12) (1 − c ) | x | + c (cid:12)(cid:12) . Since this relation holds whenever | x | ≤ c = 1 and hence h is the identity map. Proof . [of Theorem 4.1] Recall that x ∈ E is said to be triangular if x x = x . We denote by T the set of triangular points. Note that x is triangular if and only if Ψ( ., x ) is a constant map.It follows that if x ∈ T and υ ∈ Aut D then the mapsΨ( ., υ · x ) = υ ◦ Ψ( ., x ) , Ψ( ., x · υ ) = Ψ( ., x ) ◦ υ are constant, and hence υ · x, x · υ ∈ T . It is clear that T is invariant under F and under L υ , R υ for any υ ∈ Aut D , and so T is invariant under the group G given by equation (4.2). Moreover,if x ∈ T and we define(4.4) υ ( z ) = z − x ¯ x z − , χ ( z ) = z − ¯ x x z − υ · x · χ = O . Hence T is the G -orbit of O in E .Let V denote the orbit of O under Aut E . Clearly V ⊃ T . By [11, Satz 1], V is a closedconnected complex submanifold of E . Since E is inhomogeneous, V = E . If V is a 3-dimensionalsubmanifold of E then V is both open and closed in E , and so by connectedness V = E , acontradiction. Thus V is a connected 2-dimensional submanifold of E . Since T is closed in E itfollows that T is an open and closed subset of V , hence by connectedness is equal to V . That is, T is the orbit of O under Aut E . Hence every automorphism of E restricts to an automorphismof T .Consider any f ∈ Aut E . Let f ( O ) = x , so that x ∈ T . Define υ, χ as in equations (4.4)and let g ( . ) = υ · f ( . ) · χ . Then g ∈ Aut E and g ( O ) = O . The restriction g T of g to T is anautomorphism of T . Since T is isomorphic to the bidisc D , g T induces an automorphism of D that fixes (0 , g T is one of the automorphisms( x , x , x x ) ( ωx , ηx , ωηx x ) or ( x , x , x x ) ( ηx , ωx , ωηx x )for some ω, η ∈ T . In the former case let h ( . ) = ρ ¯ ω · g ( . ) · ρ ¯ η : then h ∈ Aut E and h fixes T pointwise. By Lemma 4.2, h is the identity map id E . Thus g = ρ ω · id E · ρ η and so f = L υ R χ g = L υ ◦ ρ ω R ρ η ◦ χ ∈ G. In the latter case a similar argument shows that f = L υ ◦ ρ η R ρ ω ◦ χ F ∈ G. Thus in either case f ∈ G .5. The action of
Aut E on a foliation Condition (v) of Theorem 1.1 shows that if | β | + | β | < ϕ β β : D → C : λ ( β + ¯ β λ, β + ¯ β λ, λ )maps D into E , and moreover every point of E lies on some disc ϕ β β ( D ). If x = ϕ β β ( λ ) thenwe find that β = x − ¯ x λ − | λ | , β = x − ¯ x λ − | λ | , and so x lies in a unique disc ϕ β β ( D ). Thus the discs ϕ β β ( D ) , | β | + | β | <
1, constitute afoliation of E by analytic discs, which we shall call the β -foliation of E . It is easily checked thatΨ( ω, . ) is an analytic left inverse of ϕ β β modulo Aut D for any ω ∈ T , and hence the leaves ofthe β -foliation are complex geodesics of E .The action of Aut E can be understood in terms of its action on the β -foliation. HE AUTOMORPHISM GROUP OF THE TETRABLOCK 11
Theorem 5.1.
Aut E permutes the leaves of the β -foliation transitively. Specifically, if x ∈ ϕ β β ( D ) and υ, χ ∈ Aut D are given by (5.1) υ ( z ) = ω z − α ¯ αz − , χ ( z ) = ζ z − θ ¯ θz − then x · χ ∈ ϕ γ γ ( D ) where (5.2) γ = β (1 − | θ | ) | − ζθβ | − | θβ | , γ = ¯ θ (1 − | β | + | β | ) − ζβ − ¯ ζ ¯ θ ¯ β | − ζθβ | − | θβ | and υ · x ∈ ϕ δ δ ( D ) where (5.3) δ = ω α (1 − | β | + | β | ) − β − α ¯ β | − ¯ αβ | − | αβ | , δ = β (1 − | α | ) | − ¯ αβ | − | αβ | . Moreover, for any β , β such that | β | + | β | < , if υ, χ in equations (5.1) are chosen with ω = ζ = 1 , α = ξ tanh { tanh − ( | β | + | β | ) + tanh − ( | β | − | β | ) } , (5.4) θ = ξ tanh { tanh − ( | β | + | β | ) − tanh − ( | β | − | β | ) } where β = | β | ξ , ¯ β = | β | ξ and ξ , ξ ∈ T , then υ − · ϕ β β ( D ) · χ − = ϕ ( D ) = { (0 , , λ ) : λ ∈ D } . Proof . A straightforward calculation shows that(5.5) ϕ β β ( λ ) · χ = ϕ γ γ ( µ )where(5.6) µ = η λ + c ¯ cλ + 1 , η = − ζ − ¯ ζ ¯ θ ¯ β − ζθβ , c = − ¯ ζ ¯ θβ − ¯ ζ ¯ θ ¯ β and γ , γ are given by equations (5.2). Similarly υ · ϕ β β ( λ ) = ϕ δ δ ( ν )where(5.7) ν = η ′ λ + c ′ ¯ c ′ λ + 1 , η ′ = − ω − α ¯ β − ¯ αβ , c ′ = − αβ − α ¯ β and δ , δ are given by equations (5.3). Since | c | < , | c ′ | < , | η | = 1 and | η ′ | = 1, both L υ and R χ map any β -leaf bijectively onto another β -leaf. As F clearly does likewise, it follows thatevery automorphism of E permutes the leaves of the β -foliation.On applying equations (5.3) and (5.2) to the case β = β = 0 we find that(5.8) υ · ϕ ( D ) · χ = ϕ ( ωα )0 ( D ) · χ = ϕ γ γ ( D )where(5.9) γ = ωα (1 − | θ | )1 − | αθ | , γ = ¯ θ (1 − | α | )1 − | αθ | . Consider β , β such that | β | + | β | <
1. Choose ω = ζ = 1 and choose α, θ according toequations (5.4); note that | α | < , | θ | < | β | + | β | <
1, and ξ | α | = α, ¯ ξ | θ | = ¯ θ .Furthermore tanh − | α | = tanh − ( | β | + | β | ) + tanh − ( | β | − | β | ) , tanh − | θ | = tanh − ( | β | + | β | ) − tanh − ( | β | − | β | ) , whencetanh − | α | + tanh − | θ | = tanh − ( | β | + | β | ) , tanh − | α | − tanh − | θ | = tanh − ( | β | − | β | ) . On taking tanh of both sides we obtain | α | + | θ | | αθ | = | β | + | β | , | α | − | θ | − | αθ | = | β | − | β | , and therefore β = ξ | β | = ξ | α | (1 − | θ | )1 − | αθ | = α (1 − | θ | )1 − | αθ | = γ β = ¯ ξ | β | = ¯ ξ | θ | (1 − | α | )1 − | αθ | = ¯ θ (1 − | α | )1 − | αθ | = γ . Thus υ · ϕ ( D ) · χ = ϕ β β ( D ), as required. It follows that the action of Aut E on the set of β -leaves is transitive.The theorem shows that the orbit of any point of E under Aut E contains a point of the form(0 , , λ ) with λ ∈ D ; the application of a further rotation shows that we may take 0 ≤ λ < Theorem 5.2.
Let x ∈ E . The orbit of x under Aut E contains a unique point of the form (0 , , r ) with r ∈ [0 , . If x = ( β + ¯ β λ, β + ¯ β λ, λ ) then r is given by r = (cid:12)(cid:12)(cid:12)(cid:12) λ − α ¯ θ ¯ αθλ − (cid:12)(cid:12)(cid:12)(cid:12) where α, θ are given by equations (5.4). Proof . As in equations (5.8) and (5.9) we have, for υ, χ given by equations (5.1) and z ∈ D , υ · ϕ ( z ) · χ = ϕ ( ωα )0 ( ν ) · χ = ϕ γ γ ( µ )where (by equations (5.6), (5.7)) ν = − ωz, µ = ωζ z + ¯ ζα ¯ θζ ¯ αθz + 1 . Now choose ω = ζ = 1 and choose α, θ as in equations (5.4). As we showed above, γ = β and γ = β . Thus υ · (0 , , z ) · χ = ϕ β β (cid:18) z + α ¯ θ ¯ αθz + 1 (cid:19) . Substitute z = ( λ − ¯ αθ ) / ( − α ¯ θλ + 1) and apply a suitable rotation ρ to obtain υ · ρ · (0 , , r ) · χ = ϕ β β ( λ )with r as in the theorem.It remains to prove the uniqueness of r . Suppose that (0 , , r ) , (0 , , s ) both lie in the orbitof x with 0 ≤ r, s <
1; then there exist υ, χ ∈ Aut D such that υ · (0 , , r ) = (0 , , s ) · χ (we canignore F here since both points are fixed by F ). That is, if υ, χ are given by equations (5.1), ϕ ( ωα )0 ( − ωr ) = ϕ θ ( − ζs ). It follows that α = θ = 0 and ωr = ζs . Since r, s ≥ r = s .In [1, Theorem 3.4.4] it is shown by a different method that, for every x ∈ E , there exist υ, χ ∈ Aut D such that υ · x · χ = (0 , , r ) for some r ∈ [0 , υ, χ and r areobtained. HE AUTOMORPHISM GROUP OF THE TETRABLOCK 13 Concluding remarks
The original purpose for the study of both the tetrablock and the symmetrised bidisc G wasto try and solve special cases of the µ -synthesis problem ([2, Section 9], [1, 3, 4]) which is arefinement of classically-studied interpolation problems. Although the approach has indeed ledto some new results which are relevant to the motivating engineering problem, the results sofar are too special to be of great import in applications. It is reasonable to hope that a betterunderstanding of the complex geometry of these domains and analogous ones will in the futureprovide results that will be very useful for the theory of H ∞ control. Meanwhile, the studyof G has proved to be of considerable interest to specialists in several complex variables (e.g.[6, 10]; numerous authors have developed the theory of G and its higher-dimensional analoguesfurther). The appeal of these domains is that they admit a rich and explicit function theorythat is in some ways close to that of classical domains, such as Cartan domains, but in othershas new and subtle features. The present paper is more “several complex variables” than “ H ∞ control”: it addresses some of the basic questions one would ask about any domain of interest.It does however have some implications for cases of the µ -synthesis problem. For example,suppose we are given 2 × A, B with A = [ A A ] strictly triangular (butnot 0) and B = [ B B ] = [ b ij ] not diagonal, and we are asked whether there exists an analyticmatrix function F in the unit disc such that F (0) = A, F ′ (0) = B and µ ( F ( λ )) ≤ λ ∈ D (here µ is a certain cost function lying between the spectral radius and the operatornorm; see [2, Section 9]). It follows from Theorem 2.1 above that such an F exists if and only ifmax {| b | , | b |} + | A ∧ B + A ∧ B | ≤
1. Such explicit criteria for µ -synthesis problems arehard to come by in general. Again, knowledge of the automorphisms of E reveals a non-obviousequivalence between certain µ -synthesis problems. References [1] A. A. Abouhajar,
Function theory related to H ∞ control, Ph.D. thesis, Newcastle University, 2007.[2] A. A. Abouhajar, M. C. White and N. J. Young, A Schwarz lemma for a domain related to mu-synthesis,arXiv:0708.0637.[3] J. Agler and N. J. Young, The two-by-two spectral Nevanlinna-Pick problem,
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