The Lempert function of the symmetrized polydisc in higher dimensions is not a distance

Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc

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.

Kähler submanifolds of 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.

$L^p$正则性研究：伯格曼投影在对称化多边形域上的表现

$L^p$ 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。

Nonisotropically balanced domains, Lempert function estimates, and the spectral Nevanlinna-Pick problem

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…

Characterizations of Symmetrized Polydisc

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 $|p| \leq 1$.

Characterizations of the symmetrized polydisc via another family of domains

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 = …

A Schwarz lemma for the symmetrized polydisc via estimates on another family of domains

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…

对称多圆盘的函数论性质及其推广

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$\boldsymbol{\theta_d}$。

A Schwarz lemma for two families of domains and complex geometry

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 $\widetildeΓ_n$.

Rational dilation for operators associated with spectral interpolation and distinguished varieties

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.

Green functions of the spectral ball and symmetrized polydisk

The Green function of the spectral ball is constant over the isospectral varieties, is never less than the pullback of its counterpart on the symmetrized polydisk, and is equal to it in the generic case where the pole is a cyclic (non-derogatory) matrix. When the pole is derogatory, the inequality is always strict, and the difference between the two functions depends on the order of nilpotence of the strictly upper triangular blocks that appear in the Jordan decomposition of the pole. In particular, the Green function of the spectral ball is not symmetric in its arguments. Additionally, some e…

Lifting maps from the symmetrized polydisk in small dimensions

The spectral unit ball $Ω_n$ is the set of all $n\times n$ matrices with spectral radius less than $1$. Let $π(M) \in \mathbb C^n$ stand for the coefficients of its characteristic polynomial of $M$ (up to signs), i.e. the elementary symmetric functions of its eigenvalues. The symmetrized polydisk is $\mathbb G_n:=π(Ω_n)$. When investigating Nevanlinna-Pick problems for maps from the disk to the spectral ball, it is often useful to project the map to the symmetrized polydisk (for instance to obtain continuity results for the Lempert function): if $ψ\in \mathcal O(\mathbb D, Ω_n)$, then $π\circ …

On decomposition of operators having $Γ_3$ as a spectral set

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…