Isabelle Chalendar, Pamela Gorkin, Jonathan R. Partington
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Singular inner eigenfunctions of composition operators
本文针对单位圆盘自同构φ产生的复合算子C_φ，刻画其源于离散测度的奇异内本征函数，建立Beurling与模型不变子空间的联系完成内函数分类，求解本征函数方程并给出该联系的应用。
Volterra-type inner derivations on Hardy spaces
该研究针对巴拿赫空间上不满足Calkin经典内导子结论的情况，借助广义面积算子与算子紧缠绕关系，刻画了Hardy空间上的Volterra型内导子及Hardy与Bergman空间间算子的紧缠绕关系，补充了Calkin的结果。
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。
We introduce another new type of combinations of Bernstein operators in this paper, which can be used to approximate the functions with inner singularities. The direct and inverse results of the weighted approximation of this new type combinations are obtained.
We introduce a new type of Bernstein operators, which can be used to approximate the functions with inner singularities. The direct and inverse results of the weighted approximation of this new type of combinations are given.
We consider the pointwise weighted approximation by Bernstein operators with inner singularities. The related weight functions are weights $\bar{w}(x)=|x-ξ|^α(0<ξ<1,\ α>0).$ In this paper we give direct and inverse results of this type of Bernstein polynomials.
In this work, generalizations of some inequalities for continuous $h$-synchronous ($h$-asynchronous) functions of linear bounded selfadjoint operators under positive linear maps in Hilbert spaces are proved.
In this paper, we propose to define the concept of family of local atoms and then we generalize this concept to the atomic system for operator in Banach spaces by using semi-inner product. We also give a characterization of atomic systems leading to obtain new frames. In addition, a reconstruction formula is obtain. Next, some new results are established. The characterization of atomic systems allows us to state some results for sampling theory in semi-inner product reproducing kernel Banach spaces. Finally, having used frame operator in Banach spaces, new perturbation results are established.
We obtain a complete characterization of the norm attainment set of a bounded linear functional on a normed space, in terms of a semi-inner-product defined on the space. Motivated by this result, we further apply the concept of semi-inner-products to obtain a complete characterization of the norm attainment set of a bounded linear operator between any two real normed spaces. In particular, this answers an open question raised recently in [Sain, D., \textit{On the norm attainment set of a bounded linear operator}, J. Math. Anal. Appl., \textbf{457} (2018), 67-76.]. Our results illustrate the ap…
Let ${\cal H}$ be a Hilbert space, $A$ a positive definite operator in ${\cal H}$ and $\langle f,g\rangle_A=\langle Af,g\rangle$, $f,g\in {\cal H}$, the $A$-inner product. This paper studies the geometry of the set $$ {\cal I}_A^a:=\{\hbox{ adjointable isometries for } \langle \ , \ \rangle_A\}. $$ It is proved that ${\cal I}_A^a$ is a submanifold of the Banach algebra of adjointable operators, and a homogeneous space of the group of invertible operators in ${\cal H}$, which are unitaries for the $A$-inner product. Smooth curves in ${\cal I}_A^a$ with given initial conditions, which are minima…
We introduce the notion of cascade connection of multiparametric discrete time-invariant linear dynamical systems with unit delay. This allows us to construct the explicit example of conservative realization of a decomposable operator-valued function $θ(z_1, z_2)=θ_2(z_2)θ_1(z_1)$, where $θ_1$ and $θ_2$ are inner operator-valued functions on the unit disk, vanishing at zero, and discuss some properties of such realization.
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。
In this paper, we characterize the boundedness, the compactness and the Hilbert-Schmidt property for composition operators acting from a de Branges-Rovnyak space $\mathcal H(b)$ into itself, when $b$ is a rational function in the closed unit ball of $H^\infty$ (but not a finite Blaschke product). In particular, we extend some of the results obtained by D. Sarason and J.N. Silva in the context of local Dirichlet spaces.
Inner functions play a central role in function theory and operator theory on the Hardy space over the unit disk. Motivated by recent works of C. Bénéteau et al. and of D. Seco, we discuss inner functions on more general weighted Hardy spaces and investigate a method to construct analogues of finite Blaschke products.
Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators
本文通过将内积与正算子对应，严格刻画了Hilbert–Schmidt算子空间$S_2$上的所有内积，并给出了$\mathcal B(S_2)$中算子为正的充要条件。
Given a symmetric linear transformation on a Hilbert space, a natural problem to consider is the characterization of its set of symmetric extensions. This problem is equivalent to the study of the partial isometric extensions of a fixed partial isometry. We provide a new function theoretic characterization of the set of all self-adjoint extensions of any symmetric linear transformation $B$ with equal indices and inner Livsic characteristic function $Θ_B$ by constructing a natural bijection between the set of self-adjoint extensions and the set of all contractive analytic functions $Φ$ which ar…
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 introduce the notion of spectral points of type $π_+$ and type $π_-$ of closed operators $A$ in a Hilbert space which is equipped with an indefinite inner product. It is shown that these points are stable under compact perturbations. In the second part of the paper we assume that $A$ is symmetric with respect to the indefinite inner product and prove that the growth of the resolvent of $A$ is of finite order in a neighborhood of a real spectral point of type $π_+$ or $π_-$ which is not in the interior of the spectrum of $A$. Finally, we prove that there exists a local spectral function on i…
Given a lineal H_0 and x_0\in H_0 and a linear injective operator U_0: H_0 \to H_0 such that all U_0^N, N \in {\bf Z} exist and all U_0^N x_0, N \in {\bf Z} are linearly independent, anyone can define on span{{U_0}^N x_0 | N \in {\bf Z}} a (pre)hilbert scalar product such that U_0 becomes a unitary operator. The problem under consideration is: Suppose there is specified an indefinite inner product {,}_0 on H_0 and U_0 is a {,}_0-unitary operator. Can one introduce a (pre)hilbert topology on L_{x_0} so that after completion and possible extension the resulting {,}_ext is continuous, the resulti…
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…
Let $φ$ be a function in the Hardy space $H^2(\mathbb{T}^d)$. The associated (small) Hankel operator $\mathbf{H}_φ$ is said to have minimal norm if the general lower norm bound $\|\mathbf{H}_φ\| \geq \|φ\|_{H^2(\mathbb{T}^d)}$ is attained. Minimal norm Hankel operators are natural extremal candidates for the Nehari problem. If $d=1$, then $\mathbf{H}_φ$ has minimal norm if and only if $φ$ is a constant multiple of an inner function. Constant multiples of inner functions generate minimal norm Hankel operators also when $d\geq2$, but in this case there are other possibilities as well. We investi…
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…
The numerical range of a bounded, linear operator on a Hilbert space is a set in $\mathbb{C}$ that encodes important information about the operator. In this survey paper, we first consider numerical ranges of matrices and discuss several connections with envelopes of families of curves. We then turn to the shift operator, perhaps the most important operator on the Hardy space $H^2(\mathbb{D})$, and compressions of the shift operator to model spaces, i.e.~spaces of the form $H^2 \ominus θH^2$ where $θ$ is inner. For these compressions of the shift operator, we provide a survey of results on the…
We look for an effective description of the algebra D_{Lie}(X,B) of operators on a bimodule X over an algebra B, generated by inner derivations. It is shown that in some important examples D_{Lie}(X,B) consists of all elementary operators x\to \sum_i a_ixb_i satisfying the conditions $\sum_i a_ib_i = \sum_i b_ia_i = 0. The Banach algebraic versions of these results are also obtained and applied to the description of closed Lie ideals in some Banach algebras, and to the proof of a density theorem for Lie algebras of operators on Hilbert space.
In this paper we consider generalized Hardy spaces in the octonionic setting associated to arbitrary Lipschitz domains where the unit normal field exists almost everywhere. First we discuss some basic properties and explain structural differences to the associative Clifford analysis setting. The non-associativity requires special attention in the definition of an appropriate inner product and hence in the definition of a generalized Szegö projection. Whenever we want to apply classical theorems from reproducing kernel Hilbert spaces we first need to switch to the consideration of real-valued i…
Inner product bounds via the Moore-Penrose inverse with applications
该研究针对希尔伯特空间算子，利用Moore-Penrose逆推导柯西-施瓦茨型内积界，并将其推广至涉及算子不同组合的数值半径-范数界，提供了新的精细结果。
Pairs of Clark Unitary Operators on the Bidisk and their Taylor Joint Spectra
本文研究双盘上模型空间K_ϕ上的压缩位移算子的Clark理论，探讨其与单变量情况的异同，并证明Clark正交算子的Taylor联合谱与内函数ϕ的水平集一致。
A real semi-inner-product space is a real vector space $\M$ equipped with a function $[.,.] : \M \times \M \to \Re$ which is linear in its first variable, strictly positive and satisfies the Schwartz inequality. It is well-known that the function $||x|| = \sqrt{[x,x]}$ defines a norm on $\M$. and vica versa, for every norm on $X$ there is a semi-inner-product satisfying this equality. A linear operator $A$ on $\M$ is called \emph{adjoint abelian with respect to $[.,.]$}, if it satisfies $[Ax,y]=[x,Ay]$ for every $x,y \in \M$. The aim of this paper is to characterize the diagonalizable adjoint …
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.
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 …
Let $H_m(\mathbb B,\mathcal D)$ be the $\mathcal D$-valued functional Hilbert space with reproducing kernel $K_m(z,w) = (1-\langle z,w\rangle)^{-m}1_{\mathcal D}$. A $K_m$-inner function is by definition an operator-valued analytic function $W: \mathbb B \rightarrow L(\mathcal E, \mathcal D)$ such that $\|Wx\|_{H_m(\mathbb B,\mathcal D)} = \|x\|$ for all $x \in \mathcal E$ and $(W\mathcal E) \perp M_z^α(W\mathcal E)$ for all $α\in \mathbb N^n \setminus \{0\}$. We show that the $K_m$-inner functions are precisely the functions of the form $W(z) = D + C \sum^m_{k=1}(1 - ZT^*)^{-k}ZB$, where $T \…
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。
We propose a unique way how to choose a new inner product in a Hilbert space with respect to which an originally non-self-adjoint operator similar to a self-adjoint operator becomes self-adjoint. Our construction is based on minimising a 'Hilbert-Schmidt distance' to the original inner product among the entire class of admissible inner products. We prove that either the minimiser exists and is unique, or it does not exist at all. In the former case we derive a system of Euler-Lagrange equations by which the optimal inner product is determined. A sufficient condition for the existence of the un…
The Segal algebra $\mathbf{S}_{0}(G)$ is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups $G$. It is a Banach space that exhibits a kernel theorem similar to the well-known Schwartz kernel theorem. Specifically, we call this characterization of the continuous linear operators from $\mathbf{S}_{0}(G_{1})$ to $\mathbf{S}'_{0}(G_{2})$ by generalized functions in $\mathbf{S}'_{0}(G_{1} \times G_{2})$ the "outer kernel theorem". The main subject of this paper is to formulate what we call the "inner kernel theorem". This is the characterization of those linear operators that…
More inner product bounds with applications
本文提出Hilbert空间中内积的新界，并应用于数值半径和算子范数的计算。
Which hyponormal block Toeplitz operators are either normal or analytic?
本文通过分析矩阵值内函数的最大公因子与弱交换性，将 Curto-Hwang-Lee 关于有理符号块 Toeplitz 算子的结果推广至有界型符号，证明满足标量内函数条件时，平方也次正规的次正规 T_Ψ 必为正规或解析算子。
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.
On the extension of inner derivations from dense ideals in Banach algebras
本文探讨了Banach代数中稠密理想上内导出的扩展问题，通过构造紧算子代数和有限秩算子理想，证明了所有导出到有限秩算子的理想都是内导出，但紧算子代数上存在外导出。
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。
We introduce the notions of L(H)-valued norms and Banach spaces with respect to L(H)-valued norms. In particular, we introduce Hilbert spaces with respect to L(H)-valued inner products. In addition, we provide several fundamental examples of Hilbert spaces with respect to L(H)-valued inner products.
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。
Operator-valued rational functions
本文证明了算子值坐标函数zI_E的每个内因子都是Blaschke-Potapov因子，并引入了算子值有理函数的概念，证明Δ为双侧内且有理当且仅当可表示为有限Blaschke-Potapov积，扩展了Potapov对矩阵值函数的结论。
Band projections in spaces of regular operators
该研究在Dedekind完备Banach格的正则算子空间中引入内带投影，给出原子序连续Banach格的新刻画，并明确了乘法算子成为带投影的充要条件。
In this paper we present how spectral properties of certain linear operators vary when operators are considered in different Hilbert spaces having common dense domain as the space of polynomials in one real variable with complex coefficients. This is done taking differential operator and matrix operator representation of dilation operator $E_{p,d}$ dilating $p$, a polynomial sequence, by $d$, a non-constant sequence of non-zero complex scalars. For the purpose of identification of $E_{p,d}$ with formal differential operator, which includes finite order differential operators as special cases, …
We introduce two kinds of quasi-inner functions. Since every rationally invariant subspace for a shift operator $S_K$ on a vector-valued Hardy space $H^{2}(Ω,K)$ is generated by a quasi-inner function, we also provide relationships of quasi-inner functions by comparing rationally invariant subspaces generated by them. Furthermore, we discuss fundamental properties of quasi-inner functions, and quasi-inner divisors.
For an inner function $θ$ with $θ'\in\mathcal N$, where $\mathcal N$ is the Nevanlinna class, several problems are posed in connection with the canonical (inner-outer) factorization of $θ'$.
Inner and characteristic functions in polydiscs
本文解决了多圆柱中线性算子特征函数的表示问题，给出了交换纯压缩算子组特征函数的具体描述，从而刻画了多圆柱上的内部函数。
An operator *-algebra is a non-selfadjoint operator algebra with completely isometric involution. We show that any operator *-algebra admits a faithful representation on a Hilbert space in such a way that the involution coincides with the operator adjoint up to conjugation by a symmetry. We introduce operator *-correspondences as a general class of inner product modules over operator *-algebras and prove a similar representation theorem for them. From this we derive the existence of linking operator *-algebras for operator *-correspondences. We illustrate the relevance of this class of inner p…
This paper extends three results from classical finite frame theory over real or complex numbers to binary frames for the vector space ${\mathbb Z}_2^d$. Without the notion of inner products or order, we provide an analog of the "fundamental inequality" of tight frames. In addition, we prove the binary analog of the characterization of dual frames with given inner products and of general frames with prescribed norms and frame operator.
We study Riesz bases/Riesz sequences of reproducing kernels in the model space $K_θ$ in connection with the corresponding Schur--Nevanlinna parameters and functions. In particular, we construct inner functions with given Schur--Nevanlinna parameters at a given sequence $Λ$ such that the corresponding systems of projections of reproducing kernels in the model space are complete/non complete. Furthermore, we give a compactness criterion for Hankel operators with symbol $θ\overline B$, where $θ$ is an inner function and $B$ is an interpolating Blaschke product and use this criterion to describe R…
We study generalized inner functions on a large family of Reproducing Kernel Hilbert Spaces. We show that the only inner functions that are entire are the normalized monomials.
Disjointly universal inner functions
本文刻画了两个复合算子序列何时允许不相交泛型的Blaschke乘积和奇异内函数，特征依赖于符号的几何性质如双曲导数和伪双曲距离。
We study a concept of inner function suited to Dirichlet-type spaces. We characterize Dirichlet-inner functions as those for which both the space and multiplier norms are equal to 1.
For a large class of unitarily invariant reproducing kernel functions $K$ on the unit ball $\mathbb B_d$ in $\mathbb C^d$, we characterize the $K$-inner functions on $\mathbb B_d$ as functions admitting a suitable transfer function realization. We associate with each $K$-contraction $T \in L(H)^d$ a canonical operator-valued $K$-inner function and extend a uniqueness theorem of Arveson for minimal $K$-dilations to our setting. We thus generalize results of Olofsson for $m$-hypercontractions on the unit disc and of the first named author for $m$-hypercontractions on the unit ball.
We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner 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。
For a matrix-valued function $Φ\in L^2_{M_{n\times m}}$, it is well-known that the kernel of a block Hankel operator $H_Φ$ is an invariant subspace for the shift operator. Thus, if the kernel is nontrivial, then $\ker H_Φ= ΘH^2_{\mathbb C^r}$ for a natural number $r$ and an $m\times r$ matrix inner function $Θ$ by Beurling-Lax-Halmos Theorem. It will be shown that the size of the matrix inner function $Θ$ associated with the kernel of a block Hankel operator $H_Φ$ is closely related with a certain independency of the columns of $Φ$, which is defined in this paper. As an important application o…
The objective of this paper is to study wandering subspaces for commuting tuples of bounded operators on Hilbert spaces. It is shown that, for a large class of analytic functional Hilbert spaces $\mathcal{H}_K$ on the unit ball in $\mathbb C^n$, wandering subspaces for restrictions of the multiplication tuple $M_z = (M_{z_1}, \ldots ,M_{z_n})$ can be described in terms of suitable $\mathcal{H}_K$-inner functions. We prove that $\mathcal{H}_K$-inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogenous polynomials as an application. Along the way w…
Normal approximation for iterated inner functions
论文通过基本转移论证和经典鞅理论结果，推导出迭代内函数线性组合的Berry-Esséen定理，并简化解析内函数中心极限定理。
Mosic and Djordjevic introduced the notation of the gDMP inverse for Hilbert space operators in [J. Spectr. Theory, 8(2):555-573, 2018] by considering generalized Drazin inverse with the Moore-Penrose inverse. This paper introduces two new classes of inverses: GD1 (generalized Drazin and inner) inverse and 1GD (inner and generalized Drazin) inverse for Hilbert space operators. The existence and uniqueness of the GD1 (also 1GD) inverse are discussed, along with some properties through core-quasinilpotent decomposition and closed range decomposition operator. We further establish a few explicit …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。