Peyo Stoilov
An elementary method is given for estimates of the norms of the Toeplitz operators, determined by rational inner functions
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In this paper we survey and bring together several approaches to obtaining inner functions for Toeplitz operators. These approaches include the classical definition, the Wold decomposition, the operator-valued Poisson Integral, and Clark measures. We then extend these notions somewhat to inner functions on model spaces. Along the way we present some novel examples.
Functions in backward shift invariant subspaces have nice analytic continuation properties outside the spectrum of the inner function defining the space. Inside the spectrum of the inner function, Ahern and Clark showed that under some distribution condition on the zeros and the singular measure of the inner function, it is possible to obtain non-tangential boundary values of every function in the backward shift invariant subspace as well as for their derivatives up to a certain order. Here we will investigate, at least when the inner function is a Blaschke product, the non-tangential boundary…
We consider the asymptotic behavior of the eigenvalues of Toeplitz matrices with rational symbol as the size of the matrix goes to infinity. Our main result is that the weak limit of the normalized eigenvalue counting measure is a particular component of the unique solution to a vector equilibrium problem. Moreover, we show that the other components describe the limiting behavior of certain generalized eigenvalues. In this way, we generalize the recent results of Duits and Kuijlaars for banded Toeplitz matrices.
Composition operators between Toeplitz kernels
本文研究Toeplitz核在内函数复合算子下的像，确定包含该像的最小Toeplitz核，推广至加权情形，并给出最小模型空间对应结果及显式极大向量。
A new approach to problems of the Uncertainty Principle in Harmonic Analysis, based on the use of Toeplitz operators, has brought progress to some of the classical problems in the area. The goal of this paper is to develop and systematize the function theoretic component of the Toeplitz approach by introducing a partial order on the set of inner functions induced by the action of Toeplitz operators. We study connections of the new order with some of the classical problems and known results. We discuss remaining problems and possible directions for further research.
Two-Variable Compressions of Shifts, Toeplitz Operators, and Numerical Ranges
本文研究双变量压缩的移位、托普利兹算子和数值范围，证明有理内函数由托普利兹算子符号决定，但数值范围不一定决定其唯一性。
We prove the spectral mapping theorem $σ_e(A_ϕ) = ϕ(σ_e(A_z))$ for the Fredholm spectrum of a truncated Toeplitz operator $A_ϕ$ with symbol $ϕ$ in the Sarason algebra $C+H^\infty$ acting on a coinvariant subspace $K_θ$ of the Hardy space $H^2$. Our second result says that a truncated Hankel operator on the subspace $K_θ$ generated by a one-component inner function $θ$ is compact if and only if it has a continuous symbol. We also suppose a description of truncated Toeplitz and Hankel operators in Schatten classes $S^p$.
This tutorial paper presents a survey of results, both classical and new, linking inner functions and operator theory. Topics discussed include invariant subspaces, universal operators, Hankel and Toeplitz operators, model spaces, truncated Toeplitz operators, restricted shifts, numerical ranges, and interpolation.
A truncated Toeplitz operator is the compression $A_ϕ:\K_Θ \to \K_Θ$ of a Toeplitz operator $T_ϕ:H^2\to H^2$ to a model space $\K_Θ := H^2 \ominus ΘH^2$. For $Θ$ inner, let $\T_Θ$ denote the set of all bounded truncated Toeplitz operators on $\K_Θ$. Our main result is a necessary and sufficient condition on inner functions $Θ_1$ and $Θ_2$ which guarantees that $\mathcal{T}_{Θ_1}$ and $\mathcal{T}_{Θ_2}$ are spatially isomorphic (i.e., $U\T_{Θ_1} = \T_{Θ_2}U$ for some unitary $U:\K_{Θ_1} \to \K_{Θ_2}$). We also study operators which are unitarily equivalent to truncated Toeplitz operators and w…
Inner approximations of doubling weights with applications to Beurling-Malliavin theory in Toeplitz kernels
本文研究了如何用Meromorphic内函数的论证逼近任意递增函数，特别探讨了满足双倍条件的函数的逼近问题，并应用于Toeplitz核零集的密度条件和模型空间的Beurling-Malliavin主量类描述。
Partially isometric truncated and dual truncated Toeplitz operators
本文研究了符号形式为ϕ=\overline{u}v的截断Toeplitz算子和对偶截断Toeplitz算子的部分等距性质，并获得了关于非零部分等距截断和对偶截断Toeplitz算子极向量空间的若干特征结果。
A Toeplitz operator $T_φ$, $φ\in L^\infty(\mathbb{T}^n)$, is a partial isometry if and only if there exist inner functions $φ_1, φ_2 \in H^\infty(\mathbb{D}^n)$ such that $φ_1$ and $φ_2$ depends on different variables and $φ= \barφ_1 φ_2$. In particular, for $n=1$, along with new proof, this recovers a classical theorem of Brown and Douglas. \noindent We also prove that a partially isometric Toeplitz operator is hyponormal if and only if the corresponding symbol is an inner function in $H^\infty(\mathbb{D}^n)$. Moreover, partially isometric Toeplitz operators are always power partial isometry …
Multipliers between kernels of Toeplitz operators are characterised in terms of test functions (so-called maximal vectors for the kernels); these maximal vectors may easily be parametrised in terms of inner and outer factorizations. Immediate applications to model spaces are derived. The case of surjective multipliers is also analysed. These ideas are applied to describing equivalences between two Toeplitz kernels.
We study hypercyclicity of Toeplitz operators in the Hardy space $H^2(\mathbb{D})$ with symbols of the form $R(\overline{z}) +ϕ(z)$, where $R$ is a rational function and $ϕ\in H^\infty(\mathbb{D})$. We relate this problem to cyclicity of certain families of functions for analytic Toeplitz operators and give new sufficient conditions for hypercyclicity based on deep results of B. Solomyak.
We investigate truncated Toeplitz operators belonging to the Schatten ideals. We completely characterize such operators when they have an analytic symbol or belong to the ideal of Hilbert-Schmidt operators. We also study model spaces generated by Blaschke products associated with thin sequences, model spaces generated by certain types of singular inner functions, and operators associated with a class of very smooth symbols.
We apply the theory of de Branges-Rovnyak spaces to describe kernels of some Toeplitz operators on the classical Hardy space $H^2$. In particular, we discuss the kernels of the operators $T_{\bar f/ f}$ and $T_{\bar I\bar f/ f}$, where $f$ is an outer function in $H^2$ and $I$ is inner such that $I(0)=0$. We also obtain results on de Branges-Rovnyak spaces generated by nonextreme functions.
Paired and Toeplitz + Hankel operators
本文研究了向量值Hardy空间上Toeplitz+Hankel算子的完整分类，并通过内函数视角分析了$L^2(\mathbb{T})$上配对算子的性质。
We consider the family of Toeplitz operators $T_{J\bar S^{a}}$ acting in the Hardy space $H^2$ in the upper halfplane; $J$ and $S$ are given meromorphic inner functions, and $a$ is a real parameter. In the case where the argument of $S$ has a power law type behavior on the real line, we compute the critical value $$ c(J,S)=\inf\left\{a: \ker T_{J\bar S^{a}}\ne0\right\}.$$ The formula for $ c(J,S)$ generalizes the Beurling-Malliavin theorem on the radius of completeness for a system of exponentials.
We analyze certain compositions of rational inner functions in the unit polydisk $\mathbb{D}^{d}$ with polydegree $(n,1)$, $n\in \mathbb{N}^{d-1}$, and isolated singularities in $\mathbb{T}^d$. Provided an irreducibility condition is met, such a composition is shown to be a rational inner function with singularities in precisely the same location as those of the initial function, and with quantitatively controlled properties. As an application, we answer a $d$-dimensional version of a question posed in \cite{BPS22} in the affirmative.
We establish the existence of a finite-dimensional unitary realization for every matrix-valued rational inner function from the Schur--Agler class on a unit square-matrix polyball. In the scalar-valued case, we characterize the denominators of these functions. We also show that every polynomial with no zeros in the closed domain is such a denominator. One of our tools is the Korányi--Vagi theorem generalizing Rudin's description of rational inner functions to the case of bounded symmetric domains; we provide a short elementary proof of this theorem suitable in our setting.
Composition operators and Rational Inner Functions on the bidisc
本文研究双圆盘上有理内函数诱导的复合算子，证明在温和条件下其无界性，并在多项式稳定条件下证明其在不同Bergman空间之间的有界性。
In this note, we give a description of rational maps from the open unit disc $\mathbb{D}$ to the pentablock that map the boundary of $\mathbb{D}$ to the distinguished boundary of the pentablock. We also obtain a new characterization of the distinguished boundary of the pentablock.
We study membership of rational inner functions in Dirichlet-type spaces in polydisks. In particular, we prove a theorem relating such inclusions to $H^p$ integrability of partial derivatives of a RIF, and as a corollary we prove that all rational inner functions on $\mathbb{D}^n$ belong to $\mathcal{D}_{1/n, \ldots ,1/n}(\mathbb{D}^n)$. Furthermore, we show that if $1/p \in \mathcal{D}_{α,...,α}$, then the RIF $\tilde{p}/p \in \mathcal{D}_{α+2/n,...,α+2/n}$. Finally we illustrate how these results can be applied through several examples, and how the Lojasiewicz inequality can sometimes be app…
Composition Operators and Rational Inner Functions II: Boundedness between two different Bergman Spaces
本文研究非光滑有理内函数诱导的复合算子在两个不同Bergman空间间的有界性，给出了对应参数条件下的一个充分条件。
On products and partial isometry of Toeplitz operators with operator-valued symbols
本文刻画了算子值符号Toeplitz算子乘积仍为Toeplitz算子的条件，并证明了部分等距Toeplitz算子可分解为两个算子值内函数对应的乘子及其伴随的乘积，推广了标量值Hardy空间上的经典结果。
On the membership of two-variable Rational Inner Functions in spaces of Dirichlet-type
研究二变量有理内函数在狄利克雷空间及高阶变体中的成员资格，通过接触阶概念进行刻画并探讨其应用。
Let \[ Γ= \{(z+w, zw): |z|\leq 1, |w|\leq 1\} \subset \mathbb{C}^2. \] A $Γ$-inner function is defined to be a holomorphic map $h$ from the unit disc $\mathbb{D}$ to $Γ$ whose boundary values at almost all points of the unit circle $\mathbb{T}$ belong to the distinguished boundary $bΓ$ of $Γ$. A rational $Γ$-inner function $h$ induces a continuous map $h|_\mathbb{T}$ from the unit circle to $bΓ$. The latter set is topologically a Möbius band and so has fundamental group $\mathbb{Z}$. The {\em degree} of $h$ is defined to be the topological degree of $h|_\mathbb{T}$. In a previous paper the aut…
On the growth of resolvent of Toeplitz operators
本文研究 Hardy 空间上 Toeplitz 算子预解式随到谱距离的增长，证明 Laurent 多项式符号下为二次增长、附加条件下为线性增长，并推广至一类非有理符号。
The set \[ Γ{\stackrel{\rm def}{=}} \{(z+w,zw):|z|\leq 1,|w|\leq 1\} \subset {\mathbb{C}}^2 \] has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the Möbius band and it has a special subvariety which is the only complex geodesic that is invariant under all automorphisms. We exploit this geometry to develop an explicit and detailed structure theory for the rational maps from the unit disc to $Γ$ that map the boundary of the disc to the distinguished boundary of $Γ$.
We give an explicit representation of central measures corresponding to finite Toeplitz non-negative definite sequences of complex $q\times q$ matrices. Such measures are intimately connected to central $q\times q$ Carathéodory functions. This enables us to prove an explicit representation of the non-stochastic spectral measure of an arbitrary multivariate autoregressive stationary sequence in terms of the covariance sequence.
We prove two results with regard to rational inner functions in the Schur-Agler class of the tridisk. Every rational inner function of degree (n,1,1) is in the Schur-Agler class, and every rational inner function of degree (n,m,1) is in the Schur-Agler class after multiplication by a monomial of sufficiently high degree.
A note on homotopies of rational matrix inner functions
本文证明了当m>n时，圆盘内的m×n矩阵值有理内函数空间是路径连通的。
Higher-dimensional singularities of rational inner functions
本文研究多变量有理内函数的高维奇点，证明导数可积性条件沿零集分量不变且垂直直线奇点不影响全局可积性，并推广至n变量，给出新例。
Rational inner functions on the polydisk -- a survey
本文综述多圆盘上的有理内函数，重点整理二元情形下与插值、平方和公式及边界行为相关的结果，并在末节讨论高维推广。
Interpolation by Rational $Γ_{E(3; 3; 1, 1, 1)}$-Inner Function and Rational $Γ_{E(3; 2; 1, 2)}$-Inner Function
该研究探究两类特殊域的函数论性质，引入两类内函数概念并建立其与其他内函数的关联，构造对应有理内函数，解决相关插值问题并得到显式内插值函数。
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 sugge…
Let $S$ be the shift operator on the Hardy space $H^2$ and let $S^*$ be its adjoint. A closed subspace $\FF$ of $H^2$ is said to be nearly $S^*$-invariant if every element $f\in\FF$ with $f(0)=0$ satisfies $S^*f\in\FF$. In particular, the kernels of Toeplitz operators are nearly $S^*$-invariant subspaces. Hitt gave the description of these subspaces. They are of the form $\FF=g (H^2\ominus u H^2)$ with $g\in H^2$ and $u$ inner, $u(0)=0$. A very particular fact is that the operator of multiplication by $g$ acts as an isometry on $H^2\ominus uH^2$. Sarason obtained a characterization of the func…
Composition operators and Rational Inner Functions on the bidisc: A geometric approach
研究复合算子在双变元加权伯格曼空间上的性质，通过几何方法探讨有理内函数诱导算子的有界性条件，扩展了Bayart-Kosiński的结果。
Every rational Nevanlinna function in n variables is a Cayley inner function in n + 1 variables with one variable fixed in the upper half-plane.
Clark Measures Associated with Rational Inner Functions on Bounded Symmetric Domains
本文研究了有界对称域上与有理内函数相关的 Clark 测度，给出了其显式公式，并刻画了多圆盘上 $H^2(μ_α)=L^2(μ_α)$ 成立的条件及一般域的充要条件。
Rational $\mathbf{Θ_n}$-Inner Function and its Application in Interpolation Problem
研究区域$\mathbf{\Theta}_n$的几何与函数理论性质，引入$\mathbf{\Theta}_n$-内函数概念并建立其与其他函数的联系，推导有理$\mathbf{\Theta}_n$-内函数的刻画，还给出满足特定插值数据的该函数的显式公式。
We analyze the singularities of rational inner functions on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative $H^{\mathfrak{p}}$ membership purely in terms of contact order, a measure of the rate at which the zero set of a rational inner function approaches the distinguished boundary of the bidisk. We also show that derivatives of rational inner functions with singularities fail to be in $H^{\mathfrak{p}}$ for $\mathfrak{p}\ge\frac{3}{2}$ and that higher non-tangential regula…
For a bounded function $φ$ on the unit circle $\mathbb T$, let $T_φ$ be the associated Toeplitz operator on the Hardy space $H^2$. Assume that the kernel $$K_2(φ):=\{f\in H^2:\,T_φf=0\}$$ is nontrivial. Given a unit-norm function $f$ in $K_2(φ)$, we ask whether an identity of the form $|f|^2=\frac12\left(|f_1|^2+|f_2|^2\right)$ may hold a.e. on $\mathbb T$ for some $f_1,f_2\in K_2(φ)$, both of norm $1$ and such that $|f_1|\ne|f_2|$ on a set of positive measure. We then show that such a decomposition is possible if and only if either $f$ or $\overline{zφf}$ has a nontrivial inner factor. The pr…
In this paper we are discussing various aspects of wavelet filters. While there are earlier studies of these filters as matrix valued functions in wavelets, in signal processing, and in systems, we here expand the framework. Motivated by applications, and by bringing to bear tools from reproducing kernel theory, we point out the role of non-positive definite Hermitian inner products (negative squares), for example Krein spaces, in the study of stability questions. We focus on the non-rational case, and establish new connections with the theory of generalized Schur functions and their associate…
Clark measures for rational inner functions II: general bidegrees and higher dimensions
本文研究双圆盘二元及高维无奇性有理内函数的 Clark 测度，刻画其支撑、权重、酉嵌入条件，并揭示权重消失与奇性接触阶的关系。
We examine iteration of certain skew-products on the bidisk whose components are rational inner functions, with emphasis on simple maps of the form $Φ(z_1,z_2) = (ϕ(z_1,z_2), z_2)$. If $ϕ$ has degree $1$ in the first variable, the dynamics on each horizontal fiber can be described in terms of Möbius transformations but the global dynamics on the $2$-torus exhibit some complexity, encoded in terms of certain $\mathbb{T}^2$-symmetric polynomials. We describe the dynamical behavior of such mappings $Φ$ and give criteria for different configurations of fixed point curves and rotation belts in term…
Clark Measures for Rational Inner Functions
本文研究双圆盘上二元有理内函数对应的 Clark 测度与等距，在次数 (n,1) 情形完整刻画其支集、权重及酉嵌入算子，并联系 Agler 分解与正多重调和测度极点。
We give a characterization for the existence of a holomorphic interpolant on the unit polydisc $\mathbb{D}^n,$ $n\geq 2,$ for prescribed three-point Pick--Nevanlinna data. One of the key steps is a characterization for the existence of an interpolant that is a rational inner function on $\mathbb{D}^n.$ The latter reduces the search for a three-point interpolant to finding a single rational inner function that satisfies a type of positivity condition and arises from a polynomial of a very special form. This in turn relies on a pair of results, which are of independent interest, on the factoriza…
We analyze the behavior of rational inner functions on the unit bidisk near singularities on the distinguished boundary $\mathbb{T}^2$ using level sets. We show that the unimodular level sets of a rational inner function $ϕ$ can be parametrized with analytic curves and connect the behavior of these analytic curves to that of the zero set of $ϕ$. We apply these results to obtain a detailed description of the fine numerical stability of $ϕ$: for instance, we show that $\frac{\partial ϕ}{\partial z_1}$ and $\frac{\partial ϕ}{\partial z_2}$ always possess the same $L^{\mathfrak{p}}$-integrability …
By a famous result, functions in backward shift invariant subspaces in Hardy spaces are characterized by the fact that they admit a pseudocontinuation a.e. on $\T$. More can be said if the spectrum of the associated inner function has holes on $\T$. Then the functions of the invariant subspaces even extend analytically through these holes. We will discuss the situation in weighted backward shift invariant subspaces. The results on analytic continuation will be applied to consider some embeddings of weighted invariant subspaces into their unweighted companions. Such weighted versions of invaria…
In this paper we prove a Schwarz lemma for the pentablock. The set \[ \mathcal{P}=\{(a_{21}, \text{tr} \ A, \det A) : A=[a_{ij}]_{i,j=1}^2 \in \mathbb{B}^{2\times 2}\} \] where $\mathbb{B}^{2\times 2}$ denotes the open unit ball in the space of $2\times 2$ complex matrices, is called the pentablock. The pentablock is a bounded nonconvex domain in $\Bbb{C}^3$ which arises naturally in connection with a certain problem of $μ$-synthesis. We develop a concrete structure theory for the rational maps from the unit disc $\Bbb{D}$ to the closed pentablock $\overline{\mathcal{P}}$ that map the unit cir…
We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree $(m,n,1)$ RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular leve…
The set \[ \overline{\mathbb{E}}= \{ x \in {\mathbb{C}}^3: \quad 1-x_1 z - x_2 w + x_3 zw \neq 0 \mbox{ whenever } |z| < 1, |w| < 1 \} \] is called the tetrablock and has intriguing complex-geometric properties. It is polynomially convex, nonconvex and starlike about $0$. It has a group of automorphisms parametrised by ${\mathrm{Aut}~} {\mathbb{D}} \times {\mathrm{Aut}~} {\mathbb{D}} \times {\mathbb{Z}}_2$ and its distinguished boundary $b\overline{\mathbb{E}}$ is homeomorphic to the solid torus $\overline{\mathbb{D}} \times {\mathbb{T}}$. It has a special subvariety \[\mathcal{R}_{\mathbb{\ov…
Using the notions and tools from realization in the sense of systems theory, we establish an explicit and new realization formula for families of infinite products of rational matrix-functions of a single complex variable. Our realizations of these resulting infinite products have the following four features: 1) Our infinite product realizations are functions defined in an infinite-dimensional complex domain. 2) Starting with a realization of a single rational matrix-function $M$, we show that a resulting infinite product realization obtained from $M$ takes the form of an (infinite-dimensional…
Matrix valued inner functions on the bidisk have a number of natural subspaces of the Hardy space on the torus associated to them. We study their relationship to Agler decompositions, regularity up to the boundary, and restriction maps into one variable spaces. We give a complete description of the important spaces associated to matrix rational inner functions. The dimension of some of these spaces can be computed in a straightforward way, and this ends up having an application to the study of three variable rational inner functions. Examples are included to highlight the differences between t…
For each n,N>0 we construct a set of points x_1,...,x_M in D^n with the following property: if f is a rational inner function on D^n of degree strictly less than N and g is an analytic function mapping D^n to D that satisfies g(x_i)=f(x_i) for each i=1,...,M, then g=f on D^n. In terms of the Pick problem on D^n, our result implies that for any rational inner f of degree less than N, the Pick problem with data x_1,...,x_M and f(x_1),...,f(x_M) has a unique solution.
We consider two-variable model spaces associated to rational inner functions $Θ$ on the bidisk, which always possess canonical $z_2$-invariant subspaces $\mathcal{S}_2.$ A particularly interesting compression of the shift is the compression of multiplication by $z_1$ to $\mathcal{S}_2$, namely $ S^1_Θ:= P_{\mathcal{S}_2} M_{z_1} |_{\mathcal{S}_2}$. We show that these compressed shifts are unitarily equivalent to matrix-valued Toeplitz operators with well-behaved symbols and characterize their numerical ranges and radii. We later specialize to particularly simple rational inner functions and st…
Cyclic nearly invariant subspaces for semigroups of isometries
本文研究了由单位圆盘自同构生成的离散半群及等距的强连续乘法半群的几乎不变子空间结构，利用Toeplitz核与再生核理论，在特定外因子条件下通过模型空间刻画了循环子空间。
The multivariate Herglotz-Nevanlinna class: Rational approximation
本文研究多元Herglotz-Nevanlinna类中有界全纯函数的有理逼近问题，基于Takagi-Pfister斜特征向量构造Pade型逼近格式，证明Montessus de Ballore型收敛定理，并揭示Agler函数Taylor系数集的半代数性质。
We analyse the 3-extremal holomorphic maps from the unit disc $\mathbb{D}$ to the symmetrised bidisc $ \mathcal{G}$, defined to be the set $ \{(z+w,zw): z,w\in\mathbb{D}\}$, with a view to the complex geometry and function theory of $\mathcal{G}$. These are the maps whose restriction to any triple of distinct points in $\mathbb{D}$ yields interpolation data that are only just solvable. We find a large class of such maps; they are rational of degree at most 4. It is shown that there are two qualitatively different classes of rational $\mathcal{G}$-inner functions of degree at most 4, to be call…
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