Every two variable rational inner function on the bidisk has a special representation called a transfer function realization. It is well known and related to important ideas in operator theory that this does not extend to three or more variables on the polydisk. We study the class of rational inner functions on the polydisk which do possess a transfer function realization (the Schur-Agler class) and investigate minimality in their representations. Schur-Agler class rational inner functions in three or more variables cannot be represented in a way that is as minimal as two variables might suggest.
Fachrichtung Mathematik, Universität des Saarlandes（萨尔兰州立大学数学系）；College of Mathematics and Statistics, Center of Mathematics, Chongqing University（重庆大学数学与统计学院数学中心）
In this paper, we prove a noncommutative (nc for short) analog of Schwarz lemma for the nc Schur-Agler class and prove that the regular nc Schur-Agler class and the regular free Herglotz class are homeomorphic. Moreover, we give a characterization of regular free Herglotz functions. As an application, we will show that any regular free Herglotz functions can uniformly be approximated by regular Herglotz free polynomials.
Given a domain $Ω$ in $\mathbb{C}^m$, and a finite set of points $z_1,\ldots, z_n\in Ω$ and $w_1,\ldots, w_n\in \mathbb{D}$ (the open unit disc in the complex plane), the $Pick\, interpolation\, problem$ asks when there is a holomorphic function $f:Ω\rightarrow \overline{\mathbb{D}}$ such that $f(z_i)=w_i,1\leq i\leq n$. Pick gave a condition on the data $\{z_i, w_i:1\leq i\leq n\}$ for such an $interpolant$ to exist if $Ω=\mathbb{D}$. Nevanlinna characterized all possible functions $f$ that $interpolate$ the data. We generalize Nevanlinna's result to an arbitrary set $Ω$. In this case, the fu…
The Schur class, denoted by $\mathcal{S}(\mathbb{D})$, is the set of all functions analytic and bounded by one in modulus in the open unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$, that is \[ \mathcal{S}(\mathbb{D}) = \{φ\in H^\infty(\mathbb{D}): \|φ\|_{\infty} := \sup_{z \in \mathbb{D}} |φ(z)| \leq 1\}. \] The elements of $\mathcal{S}(\mathbb{D})$ are called Schur functions. A classical result going back to I. Schur states: A function $φ: \mathbb{D} \rightarrow \mathbb{C}$ is in $\mathcal{S}(\mathbb{D})$ if and only if there exist a Hilbert space $\mathcal{H}$ and an isometry (know…
Test Functions, Kernels, Realizations and Interpolation
Jim Agler用他的实现定理彻底改变了Pick插值的领域，该定理现在被称为$\mathbb C^d$中单位球的Agler-Schur类。我们讨论了这些结果所产生的检验函数代数和对偶概念的家庭再生核，以及相关的内插定理。在使用检验函数时，理想情况下希望使用尽可能小的集合。然而，在某些情况下，检验函数的无限集合是不可避免的。在这种情况下，某些拓扑考虑就显得尤为重要。我们用例子来说明这一点，包括环形和无限多圆盘的乘子代数。
We study representations of inner functions on the bidisc from a fractional linear transformation point of view, and provide sufficient conditions, in terms of colligation matrices, for the existence of two-variable inner functions. Here the sufficient conditions are not necessary in general, and we prove a weak converse for rational inner functions that admit one variable factorization. We present a complete classification of de Branges-Rovnyak kernels on the bidisc (which equally works in the setting of polydisc and the open unit ball of $\mathbb{C}^n$, $n \geq 1$). We also classify, in term…
Given a domain $Ω$ in $\mathbb{C}^n$ and a collection of test functions $Ψ$ on $Ω$, we consider the complex-valued $Ψ$-Schur-Agler class associated to the pair $(Ω,\,Ψ)$. In this article, we characterize interpolating sequences for the associated Banach algebra of which the $Ψ$-Schur-Agler class is the closed unit ball. When $Ω$ is the unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$ and the class of test function includes only the identity function on $\mathbb{D}$, the aforementioned algebra is the algebra of bounded holomorphic functions on $\mathbb{D}$ and in this case, our characte…
On the Bessmertnyi Class of Homogeneous Positive Holomorphic Functions of Several Variables
研究了开右多半平面上的一次齐次全纯算子值函数类，其实部为半正定的实部，实点上的算子值为自伴的算子值函数，以及由算子系数为半正定的线性算子束的块的Schur补表示的函数构成的子类.后一类是M. F. Bessmertny \u给出了广义Bessmertny \u类的几个等价刻画，并建立了它与单位多圆盘上全纯压缩算子值函数的Agler-Schur类的密切联系.
