A. Edigarian, W. Zwonek
We describe all proper holomorphic mappings of the symmetrized polydisc and study its geometric properties. We also apply the obtained results to the study of the spectral unit ball in $\MM_n(\CC^n)$.
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We prove that the symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains.
We prove that the Bergman kernel of the symmetrized polydisc in dimension greater than two has zeros.
We prove that the Lempert function of the symmetrized polydisc in dimension greater than two is not a distance.
Estimates for the Carathéodory metric on the symmetrized polydisc are obtained. It is also shown that the Carathéodory and Kobayashi distances of the symmetrized three-disc do not coincide.
Rank of the derivative of the projection to symmetrized polydisc
本文证明了从谱单位球到对称多圆盘的投影导数的秩等于求导处矩阵的最小多项式的次数。
A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions include the Szegö and the Bergman kernel on the symmetrized polydisc.
This paper proves the non-existence of common Kähler submanifolds of the complex Euclidean space and the symmetrized polydisc endowed with their canonical metrics.
$L^p$ regularity of the Bergman projection on the symmetrized polydisc
该研究探讨了伯格曼投影在对称化多边形域上的$L^p$正则性，通过分解和等价算子分析，确定了$p$的范围对$P$有界性的决定性影响。
Let $Γ_n$, $n \geq 2$, denote the symmetrized polydisc in $\mathbb{C}^n$, and $Γ_1$ be the closed unit disc in $\mathbb{C}$. We provide some characterizations of elements in $Γ_n$. In particular, an element $(s_1, \ldots, s_{n-1}, p) \in \mathbb{C}^n$ is in $Γ_n$ if and only if $s_j = β_j + \overline{β_{n-j}} p$, $j = 1, \ldots, n-1$, for some $(β_1, \ldots, β_{n-1}) \in Γ_{n-1}$, and $|p| \leq 1$.
We find new characterizations for the points in the \textit{symmetrized polydisc} $\mathbb G_n$, a family of domains associated with the spectral interpolation, defined by \[ \mathbb G_n :=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j \dots, \prod_{i=1}^n z_i \right): \,|z_i|<1, i=1,\dots,n \right \}. \] We introduce a new family of domains which we call \textit{the extended symmetrized polydisc} $\widetilde{\mathbb G}_n$, and define in the following way: \begin{align*} \widetilde{\mathbb G}_n := \Bigg\{ (y_1,\dots,y_{n-1}, q)\in \mathbb C^n :\; q \in \mathbb D, \; y_j = …
We make some sharp estimates to obtain a Schwarz lemma for the \textit{symmetrized polydisc} $\mathbb G_n$, a family of domains naturally associated with the spectral interpolation, defined by \[ \mathbb G_n :=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j \dots, \prod_{i=1}^n z_i \right): \,|z_i|<1, i=1,\dots,n \right \}. \] We first make a few estimates for the \textit{the extended symmetrized polydisc} $\widetilde{\mathbb G}_n$, a family of domains introduced in \cite{pal-roy 4} and defined in the following way: \begin{align*} \widetilde{\mathbb G}_n := \Bigg\{ (y_1,\do…
Function theoretic aspects of the symmetrized polydisc and generalization
本文研究对称多圆盘$\boldsymbol{\reals^d}$的函数论性质，引入Schur-Agler型类并建立相关定理，还将这些结果推广至更一般的对称域族$\boldsymbol{\theta_d}$。
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 $\widetildeΓ_n$.
The main aims of this article are to characterize a class of operators associated with the symmetrized polydisc that admit rational dilations on the minimal space and to show an interplay between rational dilation and distinguished varieties in the symmetrized polydisc.
We present a result on existence of some kind of peak functions for $\C$-convex domains and for the symmetrized polydisc. Then we apply the latter result to show the equivariance of the set of peak points for $A(D)$ under proper holomorphic mappings. Additionally, we present a description of the set of peak points in the class of bounded pseudoconvex Reinhardt domains.
We introduce the notion of a λ-nonisotropically balanced domain and show that the symmetrized polydisc in C^n, n \geq 2, is an example of such a domain. Given a λ-nonisotropically balanced domain Ω, we derive effective estimates from above and from below for the Lempert function at (0,z)\inΩ\timesΩ. We use these estimates to derive certain conditions for realising a two-point Nevanlinna-Pick interpolation in the symmetrized polydisc. Applying the ideas used in the derivation of our Lempert function estimates to the so-called spectral unit ball Ω_n, we deduce: a) a formula for the Lempert funct…
We prove that the Shilov boundary is invariant under proper holomorphic mappings between some classes of domains (containing among others quasi-balanced domains with the continuous Minkowski functionals). Moreover, we obtain an extension theorem for proper holomorphic mappings between quasi-circular domains. Using these results we show that there are no non-trivial proper holomorphic self-mappings in the tetrablock. Another important result of our work is a description of Shilov boundaries of a large class of domains (containing among other the symmetrized polydisc and the tetrablock). It is a…
A Product Principle for Harmonic Schwarz Lemmas: Boxes, Polydiscs, and Metric Geometry
本文建立调和映射微分范数的乘积原理，证明盒子与多圆盘上的精确算子范数估计及等号情形，并引入调和乘积度量，揭示其与Kobayashi-Royden度量的关系。
In the paper we study the geometric properties of a large family of domains, called the generalized tetrablocks, related to the $μ$-synthesis, containing both the family of the symmetrized polydiscs and the family of the $μ_{1,n}$-quotients $\mathbb E_n$, $n\geq2$, introduced recently by G. Bharali. It is proved that the generalized tetrablock cannot be exhausted by domains biholomorphic to convex ones. Moreover, it is shown that the Carathéodory distance and the Lempert function are not equal on a large subfamily of the generalized tetrablocks, containing i.a. $\mathbb E_n$, $n\geq4$. We also…
On the geometry of the symmetrized bidisc
本文研究对称双圆盘自同构群作用及其几何，证明它双全纯于 Isaev 分类中的 $\mathcal D_1$，并给出二者新刻画及相关域到双圆盘乘积的显式双全纯映射。
The symmetrized polydisc of dimension three is the set \[ Γ_3 =\{ (z_1+z_2+z_3, z_1z_2+z_2z_3+z_3z_1, z_1z_2z_3)\,:\, |z_i|\leq 1 \,,\, i=1,2,3 \} \subseteq \mathbb C^3\,. \] A triple of commuting operators for which $Γ_3$ is a spectral set is called a $Γ_3$-contraction. We show that every $Γ_3$-contraction admits a decomposition into a $Γ_3$-unitary and a completely non-unitary $Γ_3$-contraction. This decomposition parallels the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set $Γ_3$ and $Γ_3$-contr…
In this paper, we consider local holomorphic mappings f: M\to M' between real algebraic CR generic manifolds (or more generally, real algebraic sets with singularities) in the complex euclidean spaces of different dimensions and we search necessary and sufficient conditions for f to be algebraic. These conditions appear to exclude two certain flatness of M and of M'. From the point of view of CR geometry, a real analytic CR manifold M can be flat in essentially to ways, being biholomorphic to a product M_1\times Δ^k, k\geq 1, by a polydisc (algebraic degeneracy), or to a product M_1\times I^l,…
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