This paper establishes a Schwarz lemma for symmetrized polydiscs and their extensions, providing explicit interpolation methods and exploring geometric relationships within these complex domains.
Contribution
It introduces sharp estimates and explicit interpolation functions for symmetrized polydiscs, advancing the understanding of their complex geometric properties.
Findings
01
Derived sharp estimates for symmetrized polydiscs
02
Constructed explicit interpolating functions under certain conditions
03
Explored geometric relationships within families of these domains
Abstract
We make sharp estimates to obtain a Schwarz type lemma for the symmetrized polydisc \gn and for the extended symmetrized polydisc \Gn. We explicitly construct an interpolating function under certain condition. To do so, we followed the methods described in \cite{Young-LMS}. Also we find a few geometric interplay between the members of the family \Gn and its closure Γn.
π2k(B1,…,Bk)∈G2k and π2k+1(B1,…,Bk)∈G2k+1.
π2k(B1,…,Bk)∈G2k and π2k+1(B1,…,Bk)∈G2k+1.
Kn={KnoddKneven if n is odd if n is even,
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Full text
A Schwarz lemma for two families of domains and complex geometry
Sourav Pal
Mathematics Department, Indian Institute of Technology Bombay, Powai, Mumbai - 400076, India.
We make sharp estimates to obtain a Schwarz type lemma for the symmetrized polydisc Gn and for the extended symmetrized polydisc Gn. We explicitly construct an interpolating function under certain condition. To do so, we followed the methods described in [35]. Also we find a few geometric interplay between the members of the family Gn and its closure Γn.
The first author is supported by the Seed Grant of IIT Bombay, the CPDA and the INSPIRE Faculty Award (Award No. DST/INSPIRE/04/2014/001462) of DST, India. The second author is supported by a Ph.D fellowship from the University Grand Commission of India.
1. Introduction
This article is a sequel of [28]. Being motivated by the inspiring works due to Bharali, Costara, Edigarian, Kosinski, Nikolov, Zwonek [8, 14, 18, 19, 20, 21, 22, 24, 33] and few others (see references there in), we descend one more step into the depth of studying complex geometry and function theory of the symmetrized n-disk Gn for n≥3. The symmetrizedn-diskGn or simply the symmetrized polydisc, which consists of symmetric polynomials, is defined by
[TABLE]
This domain arises in the famous μ-synthesis problem, which is a part of the theory of robust control of systems comprising of interconnected electronic devices whose outputs
are linearly dependent on the inputs. Given a structureE, which is a linear subspace of Mm×n(C), the space of all m×n matrices, the functional
[TABLE]
is called a a structured singular value. If m=n and if E is the space of all scalar multiples of the identity matrix I, then μE(B) is equal to the spectral radius r(B). Also if E=Mm×n(C), then μE(B) is precisely the operator norm ∥B∥. Naturally if E is any linear subspace of Mn(C) that contains the identity matrix, then r(B)≤μE(A)≤∥B∥. For the control-theory motivations behind μE, we refer to the pioneering work of Doyle [15]. The μ-synthesis problem aims to find an analytic function f from the open unit disk D of the complex plane C to Mm×n(C) subject
to a finite number of interpolation conditions such that
μE(f(λ))<1, for all λ∈D. If E={λI:λ∈C}⊆Mn(C), then μE(B)=r(B)<1 if and only if
πn(ν1,…,νn)∈Gn (see
[14]); here ν1,…,νn are
eigenvalues of B and πn is the symmetrization map defined on Cn by
[TABLE]
where
[TABLE]
It is merely mentioned that πn is a proper holomorphic map and πn(Dn)=Gn, where Dn is the open polydisc defined by
[TABLE]
The closed symmetrized polydiscΓn, which is the closure of Gn, is given by
[TABLE]
The set Γn is polynomially convex but not convex (see [18]). It is evident from the definition that G1=D and below we provide an explicit form of G2 and G3 for the convenience of the readers.
[TABLE]
The symmetrized polydisc has attracted considerable attentions in past two decades because of its rich function theory [2, 3, 7, 14, 19, 23, 32], complex geometry [13, 18, 20, 21, 22], associated operator theory [4, 9, 10, 12, 25, 26, 29]. An interested reader can also see the articles referred there.
The classical Schwarz lemma in one variable is stated in the following way.
Theorem 1.1**.**
Let f be an analytic function on D such that ∣f(z)∣≤1, for all z∈D and f(0)=0. Then
(a)
∣f(z)∣≤∣z∣, for all z∈D,
2. (b)
∣f′(0)∣≤1.
Moreover, if ∣f′(0)∣=1 or if ∣f(z0)∣=∣z0∣ for some z0=0, then there is a constant c such that ∣c∣=1 and f(w)=cw for all w∈D.
In [28], we obtained an analogue of the first part of Theorem 1.1 for the symmetrized polydisc. The main aim of this article is to continue the same program to find an analogous part-(b) for Gn of the classical Schwarz lemma.
To study the complex geometry of Gn (and Γn) more deeply and for proving a Schwarz lemma for Gn, we introduced a new family of domains in [28], which we named extended symmetrized polydisc and defined as
[TABLE]
We called the closure of Gn, the closed extended symmetrized polydisc and denoted it by Γn. We proved in [28] that
[TABLE]
The purpose of introducing the family Gn was to make a few sharp estimates which provides a Schwarz lemma for Gn, [28]. Also we obtained a variety of characterizations for the points in Gn and Γn via a similar set of characterizations for Gn and Γn respectively, [28].
It is obvious that if (β1,…,βn−1)∈Gn−1, then ∣βj∣+∣βn−j∣<(jn). Therefore, it follows that Gn⊆Gn. In fact, G2=G2 but Gn⊊Gn for n≥3 (see [28], Lemma 3.0.2).
We introduced n−1 fractional linear transformations Φ1,…,Φn−1 and with their help we made some sharp estimates to find necessary conditions for the existence of an interpolating function from D to Gn and since Gn⊆Gn, the estimates became necessary for a Schwarz lemma for Gn (see [28]). Since the maximum modulus of each co-ordinate of a point in Gn does not exceed that of a point in Gn, these estimates are sharp for Gn too. Moreover, the functions Φ1,…,Φn−1 are specially designed for Gn and they characterize the points in Gn and Γn.
In this article, we first prove an analogue of part-(b) of Theorem 1.1 for Gn, which is Theorem 3.1 and it is one of the main results of this paper. As a consequence the desired Schwarz lemma for Gn (Theorem 3.2) follows. We also show in Theorem 3.1 that under certain condition, the achieved estimates are sufficient for the existence of an interpolating function from D to Gn. In Section 4, we explicitly construct such an interpolating function. Section 5 deals with some geometric interplay between the members of Gn and Γn. In Section 2, we accumulate few results from the literature which are used in the subsequent sections.
Note. The main idea and applied techniques to the results of Sections 3 and 4 of this article are borrowed from the paper [35], where analogous results for the tetrablock E are achieved. The primary reason for which the techniques of [35] are applicable here is that G3 is linearly isomorphic to E.
2. Background materials and preparatory results
We begin with a set of (n−1) fractional linear transformations Φ1,…,Φn−1 which we introduced in [28] to characterize the points in the extended symmetrized polydisc Gn.
Definition 2.1**.**
For z∈C, y=(y1,…,yn−1,q)∈Cn and for any j∈{1,…,n−1},
let us define
Let λ0∈D\{0} and let y0=(y10,…,yn−10,q0)∈Gn. Then in the set of following conditions, (1) implies (2) and (3).
There exists an analytic function ψ:D→Gn such that ψ(0)=(0,…,0) and ψ(λ0)=y0.
[TABLE]
*There exist [2n] number of functions F1,F2,…F[2n] in the Schur class such that
Fj(0)=[00∗0],
and Fj(λ0)=Bj, for j=1,…,[2n], where detB1=⋯=detB[2n]=q0,
yj0=(jn)[Bj]11 and yn−j0=(jn)[Bj]22.
Furthermore, if y0∈Jn then all the conditions (1)−(3) are equivalent.
The following result is known as Parrott’s Theorem and it will be used in sequel. One can see Theorem 12.22 in [34] for a proof to this result.
For any Z∈C2×2 with ∥Z∥<1, let DZ=(1−Z∗Z)21. We denote the unit ball of C2×2 by RI(2,2). Consider the following function:
[TABLE]
The function MZ is a matrix Möbius transformation that maps Z to [math]. The transformation MZ is an automorphism of RI(2,2), and (MZ)−1=M−Z.
We now present a Schwarz type lemma for Gn.
Theorem 3.1**.**
Let x=(x1,…,xn)∈Cn and there exists an analytic map ψ:D⟶Gn such that ψ(0)=(0,…,0) and ψ′(0)=x. Then
[TABLE]
The converse holds if x∈Kn.
Proof.
We already know that G2=G2 and this theorem was proved by Agler and Young for G2 (see Theorem 1.1 in [5]). For this reason we shall consider n≥3 when proving the converse part of this theorem.
Let ψ:D⟶Gn is an analytic map such that ψ(0)=(0,…,0) and ψ′(0)=x. Write ψ=(ψ1,…,ψn). Then, by Theorem 2.3, for each λ∈D∖{0} we have
[TABLE]
which is same as saying
[TABLE]
for each λ∈D∖{0}. Note that, using L-Hospital’s rule and the fact that ψ(0)=(ψ1(0),…,ψn(0))=(0,…,0), we have
[TABLE]
and
[TABLE]
So
[TABLE]
Since inequality \eqrefmax1 is true for all λ∈D∖{0}, by dividing both side of \eqrefmax1 by ∣λ∣ and letting λ⟶0, we have
[TABLE]
Since ψ′(0)=x, that is, (ψ1′(0),…,ψn′(0))=(x1,…,xn), we have
[TABLE]
We divide the converse part into two cases, n=3 and n>3.
Case-I. Suppose n=3 and condition \eqrefcondition−yn holds for x=(x1,x2,x3)∈K3, that is,
[TABLE]
We show that there exists an analytic map ψ:D⟶G3 such that ψ(0)=(0,0,0) and ψ′(0)=(x1,x2,x3). We shall follow similar technique as in Theorem 2.1 of [35] to construct such a function. We first assume that ∣x2∣≤∣x1∣. So, if x1=0, then we have x2=0 and hence from the inequality \eqrefcondition−y, we obtain ∣x3∣≤1. Now consider the function ψ(λ)=(0,0,λx3). Clearly ψ is analytic and satisfies
[TABLE]
Next, assume x1=0. According to Theorem 2.2, for any W∈S2×2 the function ψ=π3∘W=(3W11,3W22,W11W22−W12W21) is an analytic map from D to G3. Hence it is enough to show the existence of a function W=[Wij]∈S2×2 such that (π3∘W)(0)=(0,0,0) and (π3∘W)′(0)=x. Suppose W is a 2×2 matrix valued function such that
[TABLE]
where σ∈D will be chosen later. Then, ψ(0)=(π3∘W)(0)=(0,0,0) and
[TABLE]
Accordingly, ψ′(0)=x if and only if
[TABLE]
We shall find a function W∈S2×2 that satisfies equations \eqrefF(zeta) and \eqrefF′(zeta).
For any W∈S2×2, we have
[TABLE]
and
[TABLE]
For a fixed σ∈D, let
[TABLE]
Then ∥Z∥<1,
[TABLE]
Now if W∈S2×2 satisfies \eqrefF(zeta) and \eqrefF′(zeta), then W(0)=Z and hence we have
[TABLE]
Note that if W∈S2×2 then by \eqrefF(zeta) and \eqrefZ−zeta, the map
MZ∘W:D⟶RI(2,2)
satisfies the following condition
[TABLE]
Therefore, by Schwarz lemma for RI(2,2), we have ∥(MZ∘W)′(0)∥<1. Thus, if there exists a function W in S2×2 which satisfies \eqrefF(zeta) and \eqrefF′(zeta), then the matrix in the right hand side of the equation \eqrefMF′ must be a strict contraction.
Now choose σ=1−3∣x1∣. By (6), ∣x3∣<1 hence σ∈D. For a ρ∈C, define a matrix Bρ by
[TABLE]
For a fixed ρ (which is to be determined), define a function
[TABLE]
Then,
Vρ(0)=[0000] and Vρ′(0)=Bρ. We define Bρ in such a fashion because, for a suitable choice of ρ, the matrix Vρ′(0) is analogous to the matrix in equation \eqrefMF′ with above choice of σ. Since we have assumed ∣x2∣≤∣x1∣ and since condition \eqrefcondition−y holds, we have that ∣x1∣/3+∣x3∣≤1. Thus, the norm of first column of Bρ is equal to
[TABLE]
and also the norm of the second row of Bρ is
[TABLE]
Then, by Theorem 2.4, there exists a ρ∈C such that ∥Bρ∥≤1. Consequently, there exists a ρ∈C such that Vρ∈S2×2. A choice of such ρ is
[TABLE]
Now we define a function W=M−Z∘Vρ, where Z is as in \eqrefZ−zeta. Then W∈S2×2 and W(0)=M−Z(0)=Z. So W satisfies equation \eqrefF(zeta). Since Vρ(0) is the zero matrix and Vρ′(0)=Bρ, by equation \eqrefM−Z′ we have
[TABLE]
Hence the function W also satisfies equation \eqrefF′(zeta). Thus, there exists a function W∈S2×2 that satisfies \eqrefF(zeta) and \eqrefF′(zeta). The case ∣x1∣≤∣x2∣ can be dealt in similar way. Hence condition (6) is sufficient for the existence of an analytic map ψ:D⟶G3 such that ψ(0)=(0,…,0) and ψ′(0)=(x1,x2,x3).
Case-II. Let n>3. First assume that n is odd. Suppose x=(x1,…,xn)∈Kn. Then for all j=2,\dots,\Big{[}\dfrac{n}{2}\Big{]}, we have xj=n(jn)x1 and xn−j=n(jn)xn−1.
Therefore, condition (4) reduces to
[TABLE]
Hence by Case-I, there exists a function W∈S2×2 such that
[TABLE]
Now consider \Big{[}\dfrac{n}{2}\Big{]} number of 2×2 matrix valued functions W1,…,W[2n], where Wj(λ)=W(λ) for each j=1,\dots,\Big{[}\dfrac{n}{2}\Big{]}. Since n is odd, 2[2n]+1=n. So, we have
[TABLE]
and
[TABLE]
Note that, each Wj∈S2×2 and detWi=detWj for each i,j. Clearly, the function ψ=πn∘(W1,…,W[2n]) is analytic which maps D into Gn.
Now suppose n is even. In this case, x=(x1,…,xn)∈Kneven. Then, xj=n(jn)x1, xn−j=n(jn)xn−1 for all j=2,…,[2n]−1 and x[2n]=([2n]n)2nx1+xn−1. Hence condition (4) is reduced to
[TABLE]
which is same as
[TABLE]
So in a similar fashion as if n is odd, there exists a function W∈S2×2 which satisfies condition \eqrefwidehatW. Again consider 2n number of 2×2 matrix valued functions W1,…,W2n, where Wj(λ)=W(λ) for each j=1,…,2n. So, we have
[TABLE]
and
[TABLE]
The function ψ=πn∘(W1,…,W2n) is analytic and it maps D into Kneven∩Gn. Thus, for any n∈N if condition (4) holds for x∈Kn then there exists an analytic map ψ:D⟶Gn such that ψ(0)=(0,…,0) and ψ′(0)=x. The proof is now complete.
Remark. In particular if n=3, then the condition (4) is necessary and sufficient for the existence of such an interpolating function ψ for any x∈C3.
The following is a Schwarz type lemma for the symmetrized polydisc.
Theorem 3.2**.**
Let x=(x1,…,xn)∈Cn. If there exists an analytic map ψ:D⟶Gn such that ψ(0)=(0,…,0) and ψ′(0)=x, then
In Theorem 3.1, we proved the existence of an analytic function ψ mapping origin to origin and satisfying ψ′(0)=(x1,…,xn)∈Kn. In this section, we show an explicit construction of such a function ψ.
Theorem 4.1**.**
Let x∈Kn. If a function ψ=(ψ1,…,ψn) is given by
[TABLE]
when n is odd, and
[TABLE]
when n is even, where
[TABLE]
Then ψ is an analytic map from D into Gn with ψ(0)=(0,…,0) and ψ′(0)=x.
Proof.
We divide the proof into two cases, n=3 and n>3 as in Theorem 3.1. The idea and technique that are used in constructing an interpolating function ψ when n=3 are borrowed from Theorem 2.2 in [35].
Case-I: Suppose n=3. Then K3=C3. Let x∈C3 be such that \max\Bigg{\{}\dfrac{|x_{1}|}{3},\dfrac{|x_{2}|}{3}\Bigg{\}}+|x_{3}|\leq 1. For this particular case, we denote rx by lx. Then
[TABLE]
where
[TABLE]
We shall show that the function ψ given by \eqrefformulae−vp and \eqrefformulae−C is analytic, ψ(D)⊂G3, ψ(0)=(0,0,0) and ψ′(0)=x.
Suppose x1=0=x2, then ∣x3∣≤1 and lx=0. Consider the function ψ(λ)=(0,0,λx3), λ∈D. Clearly ψ is analytic, ψ(0)=(0,0,0) and ψ′(0)=(0,0,x3)=x. Now suppose ∣x2∣≤∣x1∣=0. The function ψ, given by (16) and (17), clearly satisfies ψ(0)=0. Note that
[TABLE]
Hence ψ′(0)=x.
We shall show that ψ is analytic and ψ(D)⊂G3.
Consider Z as in equation \eqrefZ−zeta with σ=1−3∣x1∣. Then
[TABLE]
Also consider Bρ as in equation \eqrefY−zeta, where ρ is given by \eqrefxi. Then ∥Bρ∥≤1 (as we observe in the proof of Theorem 3.1). Consider the function W(λ)=M−Z(λBρ), λ∈D. Then W∈S2×2. For a contraction Z, we know that Z∗DZ∗=DZZ∗ and hence DZ−1Z∗=Z∗DZ∗−1 when ∥Z∥<1. Here DZ=(1−Z∗Z)21. Thus
[TABLE]
Note that,
[TABLE]
So,
[TABLE]
Thus
[TABLE]
where
[TABLE]
From \eqrefF−lm we have
[TABLE]
Again
[TABLE]
and thus
[TABLE]
Then
[TABLE]
From the equation \eqrefF−lm, we have
[TABLE]
Consider the function
[TABLE]
Then
[TABLE]
which is of the form given in \eqrefformulae−vp. Since W∈S2×2, by Theorem 2.2 the map ψ is an analytic and ψ(D)⊂G3. The case when ∣x1∣≤∣x2∣=0, can be dealt in a similar way. Hence we are done for n=3.
Case-II: Suppose n>3. Let x=(x1,…,xn)∈Kn. First suppose n is odd. Then, xj=n(jn)x1 and xn−j=n(jn)xn−1 for all j=2,\dots,\Big{[}\dfrac{n}{2}\Big{]}. Hence,
[TABLE]
Consider y=(y1,y2,y3)=(n3x1,n3xn−1,xn).
Therefore, by hypothesis
[TABLE]
So, by Case-I, there exists W∈S2×2 such that
[TABLE]
and detW(λ)=1+λyˉ3lyλ(y3+λly), where
[TABLE]
Substituting the values of y, we get the following:
[TABLE]
Consider the 2×2 matrix valued functions W1,…,W[2n], where Wj(λ)=W(λ) for each j=1,…,[2n]. Then each Wj∈S2×2. Since n is odd, 2[2n]+1=n. Therefore,
[TABLE]
Now suppose n is even. So, [2n]=2n. In this case, x=(x1,…,xn)∈Kneven. Thus xj=n(jn)x1, xn−j=n(jn)xn−1 for all j=2,…,2n−1 and x2n=(2nn)2nx1+xn−1.
Again,
[TABLE]
Therefore, as in the case when n is odd, there exists W∈S2×2 such that (20) and (21) hold. Now consider the 2×2 matrix valued functions W1,…,W2n, where Wj(λ)=W(λ) for each j=1,…,2n. Then
[TABLE]
In both cases ψ=πn∘(W1,…,W[2n]). Clearly, the function ψ is an analytic map from D into Gn and ψ(0)=(0,…,0). If n is odd, then
[TABLE]
If n is even, we have
[TABLE]
It is evident that in either cases ψ′(0)=x and the proof is complete.
5. Geometric interplay between the members of Gn and Γn
In [28], we have witnessed several important geometric properties of Gn and Γn, e.g., Γn is polynomially convex but not convex, Gn is starlike but not circled etc. In this section, we shall see some interplay between Gn (or Γn) and Gn+1 (or, Γn+1).
Theorem 5.1**.**
Let n∈N. Suppose y=(y1,…,yn−1,q)∈Cn
(1)
*If n is an even number, then the point
*y∈Gn(*or, ∈Γn) if and only if *y^∈Gn+1(or, ∈Γn+1), where
[TABLE]
(2)
*If n is an odd number, then the point
*y∈Gn(*or, ∈Γn) if and only if *y∗∈Gn+1(or, ∈Γn+1), where
[TABLE]
Proof.
(1). First note that (jn+1)=n+1−jn+1(jn). As n is even, \Big{[}\dfrac{n}{2}\Big{]}=\dfrac{n}{2}=\Big{[}\dfrac{n+1}{2}\Big{]}. Suppose y∈Gn. Then, by Theorem 2.2, we have
[TABLE]
Consider the point y^=(y^1,…,y^n,q^), where q^=q and
Conversely, suppose y^∈Gn+1. Then for each j=1,\dots,\Big{[}\dfrac{n+1}{2}\Big{]}, we have
[TABLE]
Similarly, we have
[TABLE]
for any j=1,…,2n. Consequently, by Theorem 2.2, y∈Gn. In a similar fashion one can prove that y∈Γn if and only if y^∈Γn+1.
(2). Suppose n is odd and suppose y=(y1,…,yn−1,q)∈Gn. Then ∣q∣<1 and there exists (β1,…,βn−1)∈Cn−1 such that
[TABLE]
for each j=1,…,[2n].
Since n is odd, [2n]=2n−1 and [2n+1]=2n+1. Consider (γ1,…,γn)∈Cn, where
[TABLE]
for j=1,…,[2n].
Then, we have
[TABLE]
and
[TABLE]
for all j=1,…,[2n].
Therefore,
[TABLE]
Also
[TABLE]
and
[TABLE]
for all j=1,…,[2n].
Thus,
[TABLE]
Conversely, suppose y∗=(y1∗,…,yn∗,q)∈Gn+1. Then
[TABLE]
for all j=1,…,2n−1. By definition, there exists (γ1,…,γn)∈Cn such that
[TABLE]
for each j=1,…,[2n+1]. Consider (β1,…,βn−1)∈Cn−1, where
[TABLE]
Then
[TABLE]
for each j=1,…,[2n]. Therefore,
[TABLE]
Note that,
[TABLE]
for each j=1,…,[2n]. Therefore, y=(y1,…,yn−1,q)∈Gn. The proof of y∈Γn if and only if y^∈Γn+1 is similar.
Theorem 5.2**.**
Let y=(y1,…,yn−1,q)∈Cn.
(1)
*If n is even and *y∈Gn(*or, ∈Γn), then *yˇ∈Gn−1(or, ∈Γn−1), where
[TABLE]
(2)
*If n is odd and *y∈Gn(*or, ∈Γn), then *y~∈Gn−1(or, ∈Γn−1), where
[TABLE]
Proof.
(1). Let n∈N be even. Then, [2n−1]=2n−1. Also it is merely mentioned that (jn)=n−jn(jn−1). Now suppose y∈Gn. Then by Theorem 2.2, we have for each j\in\left\{1,\dots,\Big{[}\dfrac{n}{2}\Big{]}\right\}
[TABLE]
Consider the point
[TABLE]
Then qˇ=q and
[TABLE]
Note that for each j=1,…,2n−1, we have
[TABLE]
Hence, for any j=1,…,2n−1(=[2n−1]), we have
[TABLE]
Therefore, by Theorem 2.2, we conclude that yˇ∈Gn−1. Similarly if y∈Γn, then yˇ∈Γn−1.
(2). Suppose n is odd and let y∈Gn. Then ∣q∣<1 and there exists (β1,…,βn−1)∈Cn−1 such that
[TABLE]
for each j∈{1,…,[2n]}. Consider the given point y~=(y~1,…y~n−2,q~)∈Cn−1. Then q~=q,
[TABLE]
[TABLE]
Consider (γ1,…,γn−2)∈Cn−2, where γ2n−1=2nn+1(2β2n−1+β2n+1),
[TABLE]
Then, we have
[TABLE]
[TABLE]
for j=1,…,2n−3. Therefore,
[TABLE]
Also we have,
[TABLE]
[TABLE]
for j=1,…,2n−3. Hence, ∣q~∣<1 and there exists (γ1,…,γn−2)∈Cn−2 such that
[TABLE]
for each j=1,…,2n−1. Consequently y~=(y~1,…y~n−2,q~)∈Gn−1. The proof of y∈Γn implies y~∈Γn−1 is similar.
Theorem 5.3**.**
Let n∈N be even.
Let the point y=(y1,…,yn−1,q)∈Gn(or, ∈Γn). Then the point
[TABLE]
*and the map f:Gn→Gn+1 that maps y to y is an analytic embedding.
Let y=(y1,…,yn,q)∈Gn+1(or, ∈Γn+1). Then the point y~∈Gn(or, ∈Γn), where
[TABLE]
*The map g:Gn+1→Gn that maps y to y~ is analytic.
Let y=(y1,…,yn,q)∈Gn+1(or, ∈Γn+1). Then the point
[TABLE]
and the point
[TABLE]
where y^j=(jn+1)yj for each j. Also
the function h:Gn+1→Gn that maps y to y^♯ is analytic.
Proof.
.
Since y=(y1,…,yn−1,q)∈Gn, there exists a unique (β1,…,βn−1)∈Cn−1 such that
[TABLE]
for each j=1,…,2n. Note that y2n=β2n+βˉ2nq. Consider the given point y=(y1,…,yn,q)∈Cn+1. Then q=q,
[TABLE]
Define γj=βj and γn+1−j=βn−j for j=1,…,2n. Then,
[TABLE]
Evidently,
[TABLE]
For j=1,…,2n, we have
[TABLE]
Hence, there exists (γ1,…,γn)∈Cn such that
[TABLE]
for each j=1,…,[2n+1].
Hence y∈Gn+1. The map f is clearly an analytic embedding of Gn into Gn+1.
Let y=(y1,…,yn,q)∈Gn+1. Then, there exists (β1,…,βn)∈Cn such that
[TABLE]
for j=1,…,n. Consider the point
y~=(y~1,…,y~n−1,q~)∈Cn, where q~=q,
y~2n=2(n+1)2n+1(y2n+y2n+1) and
[TABLE]
Define (γ1,…,γn−1)∈Cn−1 as follows
[TABLE]
[TABLE]
Then for j=1,…,2n−1, we have
[TABLE]
[TABLE]
Therefore, (γ1,…,γn−1)∈Cn−1 where ∣γj∣+∣γn−j∣<(jn), for all j=1,…,2n. Also
[TABLE]
for j=1,…,2n−1. Since y2n=β2n+βˉ2n+1q and y2n+1=β2n+1+βˉ2nq, we have that
[TABLE]
Thus, there exists (γ1,…,γn−1)∈Cn−1 such that
[TABLE]
for all j=1,…,2n. Hence y~∈Gn. The map g is obviously analytic.
Let y=(y1,…,yn,q)∈Gn+1. Then there exists (β1,…,βn)∈Cn such that
[TABLE]
for j=1,…,n. Consider the point y^=(y^1,…,y^n,q^), where q^=q, and
y^j=(jn+1)yj for all j=1,…,n. Let αj=(jn+1)βj for j=1,…,n. Then
[TABLE]
Also we have
[TABLE]
for j=1,…,n. Thus, y^∈Gn+1. Next define (γ1,…,γn−1)∈Cn−1 in the following way:
[TABLE]
for j=1,…,2n−1.
Then, we have
[TABLE]
[TABLE]
Therefore, (γ1,…,γn−1)∈Cn−1 where ∣γj∣+∣γn−j∣<(jn), for all j=1,…,2n. Note that,
[TABLE]
Since y^2n=α2n+αˉ2n+1q and y^2n+1=α2n+1+αˉ2nq, we also have
[TABLE]
Hence, the point
[TABLE]
Clearly, the map h is analytic.
Theorem 5.4**.**
Let n∈N be odd.
Let the point y=(y1,…,yn−1,q)∈Gn(or, ∈Γn). Then the point
[TABLE]
*and the map f:Gn→Gn+1 that maps y to y is analytic.
Let y=(y1,…,yn,q)∈Gn+1(or, ∈Γn+1). Then the point y~∈Gn(or, ∈Γn), where
[TABLE]
*The map g:Gn+1→Gn that maps y to y~ is an analytic embedding.
Let y=(y1,…,yn,q)∈Gn+1(or, ∈Γn+1). Then the point
[TABLE]
and the point
[TABLE]
where y^j=(jn+1)yj. The map h:Gn+1→Gn that maps y to y^♯ is an analytic embedding.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
1[1] A. A. Abouhajar, M. C. White and N. J. Young, A Schwarz lemma for a domain related to μ 𝜇 \mu -synthesis, J. Geom. Anal. 17 (2007), 717 – 750.
2[2] J. Agler, Z. A. Lykova and N. J. Young, Extremal holomorphic maps and the symmetrized bidisc, Proc. Londo Math. Soc. , 106 (2013), 781 – 818.
3[3] J. Agler, Z. A. Lykova and N. J. Young, Algebraic and geometric aspects of rational Γ-inner functions, Adv. Math. , 328 (2018), 133 – 159.
4[4] J. Agler and N. J. Young, A commutant lifting theorem for a domain in ℂ 2 superscript ℂ 2 \mathbb{C}^{2} and spectral interpolation, J. Funct. Anal. 161 (1999), 452 – 477.
5[5] J. Agler and N. J. Young, A schwarz lemma for the symmetrized bidisc, Bull. London Math. Soc. , 33 (2001), 175 – 186.
6[6] J. Agler and N. J. Young, The hyperbolic geometry of the symmetrized bidisc, J. Geom. Anal. 14 (2004), no.3, 375 − 403 375 403 375-403 .
7[7] G. Bharali, Some new observations on interpolation in the spectral unit ball, Integral Equa. Oper. Theory , 59 (2007), 329 – 343.
8[8] G. Bharali, A family of domains associated with μ 𝜇 \mu -synthesis, Integral Equa. Oper. Theory 82 (2015), 267 – 285.
