Sushil Gorai, Jaydeb Sarkar
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 $|p| \leq 1$.
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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.
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 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 $\MM_n(\CC^n)$.
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.
This paper proves the non-existence of common Kähler submanifolds of the complex Euclidean space and the symmetrized polydisc endowed with their canonical metrics.
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。
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$\boldsymbol{\theta_d}$。
$L^p$ 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。
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 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…
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 $\widetildeΓ_n$.
We give a complete characterization of invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on the Hardy space $H^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$, $n >1$. In particular, this yields a complete set of unitary invariants for invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on $H^2(\mathbb{D}^n)$, $n > 1$. As a consequence, we classify a large class of $n$-tuples, $n > 1$, of commuting isometries. All of our results hold for vector-valued Hardy spaces over $\mathbb{D}^n$, $n > 1$. Our invariant subspace theorem solves the well-known open problem on cha…
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…
In this paper we prove some analogue of Wiman's type inequality for random analytic functions in the polydisc $\mathbb{D}^p=\{z\in\mathbb{C}^p\colon |z_j|<1, j\in\{1,\ldots,p\}\},\ p\in\mathbb{Z}_+$. The obtained inequality is sharp.
Let $H^2(\mathbb{D}^n)$ denote the Hardy space over the polydisc $\mathbb{D}^n$, $n \geq 2$. A closed subspace $\mathcal{Q} \subseteq H^2(\mathbb{D}^n)$ is called Beurling quotient module if there exists an inner function $θ\in H^\infty(\mathbb{D}^n)$ such that $\mathcal{Q} = H^2(\mathbb{D}^n) /θH^2(\mathbb{D}^n)$. We present a complete characterization of Beurling quotient modules of $H^2(\mathbb{D}^n)$: Let $\mathcal{Q} \subseteq H^2(\mathbb{D}^n)$ be a closed subspace, and let $C_{z_i} = P_{\mathcal{Q}} M_{z_i}|_{\mathcal{Q}}$, $i=1, \ldots, n$. Then $\mathcal{Q}$ is a Beurling quotient mod…
We obtain a complete characterization for doubly commuting mixed invariant subspaces of the Hardy space over the unit polydisc. We say a closed subspace $\mathcal{Q}$ of $H^2(\mathbb{D}^n)$ is mixed invariant if $M_{z_{j}}(\mathcal{Q}) \subseteq \mathcal{Q}$ for $1 \leq j \leq k$ and $M_{z_{j}}^*(\mathcal{Q}) \subseteq \mathcal{Q}$, $k+1 \leq j \leq n$ for some integer $k \in \{1, 2, \ldots, n-1 \}$. We prove that a mixed invariant subspace $\mathcal{Q}$ of $H^2(\mathbb{D}^n)$ is doubly commuting if and only if \[ \mathcal{Q} = ΘH^2(\mathbb{D}^k) \otimes \mathcal{Q}_{θ_1} \otimes \cdots \otime…
Isoperimetric-type inequalities for pluriharmonic functions on the polydisc
针对多圆盘上的多重调和函数，证明了Hardy空间与加权Bergman空间之间的等周型不等式，并建立了到单位球上的包含关系，常数由尖锐Riesz型估计得到。
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…
Let $\mathbb{D}^n$ be the open unit polydisc in $\mathbb{C}^n$, $n \geq 1$, and let $H^2(\mathbb{D}^n)$ be the Hardy space over $\mathbb{D}^n$. For $n\ge 3$, we show that if $θ\in H^\infty(\mathbb{D}^n)$ is an inner function, then the $n$-tuple of commuting operators $(C_{z_1}, \ldots, C_{z_n})$ on the Beurling type quotient module $\mathcal{Q}_θ$ is not essentially normal, where \[\mathcal{Q}_θ = H^2(\mathbb{D}^n)/ θH^2(\mathbb{D}^n) \quad \mbox{and} \quad C_{z_j} = P_{\mathcal{Q}_θ} M_{z_j}|_{\mathcal{Q}_θ}\quad (j = 1, \ldots, n).\] Rudin's quotient modules of $H^2(\mathbb{D}^2)$ are also s…
A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h,\]where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and $h \in \mathcal{H}$. A companion survey provides an introduction to the theory of Hilbert modules and some (Hilbert) module point of view to multivariable operator theory. The purpose of this survey is to emphasize algebraic and g…
Sharp Bohr Radii for Schwarz Functions and Directional derivative Operators in \mathbb{C}^n
本文通过Schwarz函数和方向导数算子在复空间C^n中研究多变量复变量的锐Bohr半径，推导出导数增长估计并证明所获常数的最优性。
Two problems on submodules of $H^2(\mathbb{D}^n)$
本文刻画多圆盘 Hardy 空间中可表为单变量内函数乘子之和的子模，并据此探讨 R.G. Douglas 提出的商模正交投影交换性问题。
Every real-rooted exponential polynomial is the restriction of a Lee-Yang polynomial
该研究证明所有实根指数多项式都是Lee-Yang多项式在环面正直线上的限制，结合已有结论，验证了Kurasov-Sarnak基于稳定多项式的构造可覆盖全部$\mathbb{N}$值傅里叶准晶。
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