A Note on Matrix Valued Truncated Toeplitz Operators

矩阵值截断Toeplitz算子：性质与应用

Matrix-Valued Truncated Toeplitz Operators: Properties and Applications

本文研究矩阵值截断Toeplitz算子的性质，将其应用于Nevanlinna-Pick插值问题，并刻画其酉部分、模型空间不变性及可逆性。

矩阵值非对称截断Toeplitz算子的刻画

Characterizations of Matrix Valued Asymmetric Truncated Toeplitz Operators

本文利用压缩移位、修正压缩移位及移位不变性，刻画了矩阵值非对称截断Toeplitz算子，该算子是作用于两个可能不同模型空间的乘法算子的压缩。

$C^*$-algebras generated by truncated Toeplitz operators

We obtain an analogue of Coburn's description of the Toeplitz algebra in the setting of truncated Toeplitz operators. As a byproduct, we provide several examples of complex symmetric operators which are not unitarily equivalent to truncated Toeplitz operators having continuous symbols.

Spatial isomorphisms of algebras of truncated Toeplitz operators

We examine when two maximal abelian algebras in the truncated Toeplitz operators are spatially isomorphic. This builds upon recent work of N. Sedlock, who obtained a complete description of the maximal algebras of truncated Toeplitz operators.

部分等距截断和对偶截断Toeplitz算子

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。

Product of truncated Hankel and truncated Toeplitz operators

A truncated Toeplitz operator is the compression of a classical Toeplitz operator on the Hardy space to a model space. A truncated Hankel operator is the compression of a Hankel operator on the Hardy space to the orthogonal complement of a model space. We study the product of a truncated Hankel operator and a truncated Toeplitz operator, and characterize when such a product is zero or compact.

Essentially commuting dual truncated Toeplitz operators

In this paper, we completely characterize when two dual truncated Toeplitz operators are essentially commuting and when the semicommutator of two dual truncated Toeplitz operators is compact. Our main idea is to study dual truncated Toeplitz operators via Hankel operators, Toeplitz operators and function algebras.

Unitary equivalence to a truncated Toeplitz operator: analytic symbols

Unlike Toeplitz operators on $H^2$, truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this note we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension $\leq 3$ is unitarily equivalent to a direct sum of truncated Toeplitz operators.

Recent progress on truncated Toeplitz operators

This paper is a survey on the emerging theory of truncated Toeplitz operators. We begin with a brief introduction to the subject and then highlight the many recent developments in the field since Sarason's seminal paper in 2007.

Asymmetric truncated Toeplitz operators equal to the zero operator

Asymmetric truncated Toeplitz operators are compressions of multiplication operators acting between two model spaces. These operators are natural generalizations of truncated Toeplitz operators. In this paper we describe symbols of asymmetric truncated Toeplitz operators equal to the zero operator.

Normal Truncated Toeplitz Operators

The characterization of normal truncated Toepltiz operators is first given by Chalendar and Timotin. We give an elementary proof of their result without using the algebraic properties of truncated Toeplitz operators.

Asymmetric truncated Toeplitz operators on finite-dimensional spaces

In this paper we consider asymmetric truncated Toeplitz operators acting between two finite-dimensional model spaces. We compute the dimension of the space of all such operators. We also describe the matrix representations of asymmetric truncated Toeplitz operators acting between two finite-dimensional model spaces. Our results are generalizations of the results known for truncated Toeplitz operators.

Matrix-valued truncated Toeplitz operators: unbounded symbols, kernels and equivalence after extension

This paper studies matrix-valued truncated Toeplitz operators, which are a vectorial generalisation of truncated Toeplitz operators. It is demonstrated that, although there exist matrix-valued truncated Toeplitz operators without a matrix symbol in $L^p$ for any $p \in (2, \infty ]$, there is a wide class of matrix-valued truncated Toeplitz operators which possess a matrix symbol in $L^p$ for some $p \in (2, \infty ]$. In the case when the matrix-valued truncated Toeplitz operator has a symbol in $L^p$ for some $p \in (2, \infty ]$, an approach is developed which bypasses some of the technical…

Truncated Toeplitz operators and complex symmetries

We show that truncated Toeplitz operators are characterized by a collection of complex symmetries. This was conjectured by Klis-Garlicka, Lanucha, and Ptak, and proved by them in some special cases.

Commutation relations for truncated Toeplitz operators

For truncated Toeplitz operators, which are compressions of multiplication operators to model subspaces of the Hardy space $H^2$, we obtain criteria for commutation relations. The results show an analogy to the case of Toeplitz matrices, and they extend the theory of Sedlock algebras.

Matrix valued truncated Toeplitz operators: basic properties

Matrix valued truncated Toeplitz operators act on vector-valued model spaces. They represent a generalization of block Toeplitz matrices. A characterization of these operators analogue to the scalar case is obtained, as well as the determination of the symbols that produce the zero operator.

双正交块 Toeplitz 运算符与矩阵值循环符号

Binormal block Toeplitz operators with matrix valued circulant symbols

本文研究了具有矩阵值循环符号的块 Toeplitz 运算符的双正交性，并探讨了 Toeplitz 运算符的 Γ-扩张及其不变子空间分析。

Asymmetric truncated Toeplitz operators and conjugations

Truncated Toeplitz operators in a model space are C--symmetric with respect to a natural conjugation in that space. We show that this and another conjugation associated to an orthogonal decomposition possess unique properties and we study their relations with asymmetric truncated Toeplitz operators in terms of C--symmetry. New connections with Hankel operators are established through this approach.

Truncated Toeplitz operators of finite rank

We give a complete description of the finite-rank truncated Toeplitz operators.

Recent results on truncated Toeplitz operators

Truncated Toeplitz operators are compressions of Toeplitz operators on model spaces; they have received much attention in the last years. This survey article presents several recent results, which relate boundedness, compactness, and spectra of these operators to properties of their symbols. We also connect these facts with properties of the natural embedding measures associated to these operators.

Asymmetric truncated Toeplitz operators on finite-dimensional spaces II

In this paper we present some consequences of the description of matrix representations of asymmetric truncated Toeplitz operators acting between finite-dimensional model spaces. In particular, we prove that these operators can be characterized using modified compressed shifts or the notion of shift invariance. We also describe rank-one asymmetric truncated Toeplitz operators acting between finite-dimensional model spaces. This paper is a sequel to the paper of J. Jurasik and B. Łanucha "Asymmetric truncated Toeplitz operators on finite-dimensional spaces".

Some characterizations of asymmetric truncated Toeplitz operators

It was recently proved that in some special cases asymmetric truncated Toeplitz operators can be characterized in terms of compressed shifts and rank-two operators of special form. In this paper we show that such characterizations hold in all cases.

Invertibility, Fredholmness and kernels of dual truncated Toeplitz operators

Asymmetric dual truncated Toeplitz operators acting between the orthogonal complements of two (eventually different) model spaces are introduced and studied. They are shown to be equivalent after extension to paired operators on $L^2(\mathbb T) \oplus L^2(\mathbb T)$ and, if their symbols are invertible in $L^\infty(\mathbb T)$, to asymmetric truncated Toeplitz operators with the inverse symbol. Relations with Carleson's corona theorem are also established. These results are used to study the Fredholmness, the invertibility and the spectra of various classes of dual truncated Toeplitz operator…

Symbols of compact truncated Toeplitz operators

This paper characterises the dual of the model space $K_I^1$, where $I$ is an inner function, intersected with the shifted Hardy space, $z H^1$. With this duality result, it is then shown that every bounded truncated Toeplitz operator on the model space $K_I^2$ has a bounded symbol if and only if every compact truncated Toeplitz operator on $K_I^2$ has a symbol which is of the form $I$ multiplied by a continuous function.

Characterizations of asymmetric truncated Toeplitz operators

The aim of this paper is to investigate asymmetric truncated Toeplitz operators with $L^2$ symbols between two different model spaces given by inner functions such that one divides the other. Characterizations of these operators are given in terms of rank two operators. A description of the class of symbols for which the corresponding asymmetric truncated Toeplitz operator is equal to the zero operator is also given.

A Szegö type theorem for truncated Toeplitz operators

Truncated Toeplitz operators are compressions of multiplication operators on $L^2$ to model spaces (that is, subspaces of $H^2$ which are invariant with respect to the backward shift). For this class of operators we prove certain Szegö type theorems concerning the asymptotics of their compressions to an increasing chain of finite dimensional model spaces.

Truncated Toeplitz operators and the Agler-Young class

Truncated Toeplitz operators were introduced in a seminal paper by Sarason in 2007. We show that they have an intimate connection with the Agler-Young class.

截断Toeplitz算子的有限截断

Finite sections of truncated Toeplitz operators

研究截断Toeplitz算子在模型空间上的有限截断离散化所关联的C*-代数，提出有限截断稳定性准则及谱和伪谱逼近结果。

The Norm of a Truncated Toeplitz Operator

We prove several lower bounds for the norm of a truncated Toeplitz operator and obtain a curious relationship between the $H^2$ and $H^{\infty}$ norms of functions in model spaces.

Point spectrum and hypercyclicity problem for a class of truncated Toeplitz operators

In this note we discuss an open problem whether a truncated Toeplitz operator on a model space can be hypercyclic. We compute point spectrum and eigenfunctions for a class of truncated Toeplitz operators with polynomial analytic and antianalytic parts. We show that, for a class of model spaces, truncated Toeplitz operators with symbols of the form $Φ(z) =a \bar{z} +b + cz$, $|a| \ne |c|$, have complete sets of eigenvectors, and, in particular, are not hypercyclic.

Bounded symbols and reproducing kernel thesis for truncated Toeplitz operators

Compressions of Toeplitz operators to coinvariant subspaces of $H^2$ are called truncated Toeplitz operators. We study two questions related to these operators. The first, raised by Sarason, is whether boundedness of the operator implies the existence of a bounded symbol; the second is the reproducing kernel thesis. We show that in general the answer to the first question is negative, and we exhibit some classes of spaces for which the answers to both questions are positive.

Laplace--Carleson embeddings on model spaces and boundedness of truncated Hankel and Toeplitz operators

A characterisation is given of bounded embeddings from weighted $L^2$ spaces on bounded intervals into $L^2$ spaces on the half-plane, induced by isomorphisms given by the Laplace transform onto weighted Hardy and Bergman spaces (Zen spaces). As an application necessary and sufficient conditions are given for the boundedness of truncated Hankel and Toeplitz integral operators, including the weighted case.

Asymmetric truncated Toeplitz operators and Toeplitz operators with matrix symbol

Truncated Toeplitz operators and their asymmetric versions are studied in the context of the Hardy space $H^p$ of the half-plane for $1<p<\infty$. It is shown that they are equivalent after extension to $2 \times 2$ matricial Toeplitz operators, which allows one to deduce information about their invertibility properties. Shifted model spaces are presented in the context of invariant subspaces, allowing one to deduce new Beurling--Lax theorems.

Truncated Toeplitz Operators: Spatial Isomorphism, Unitary Equivalence, and Similarity

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…

Schatten-class Truncated Toeplitz Operators

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.

Symbols of truncated Toeplitz operators

We consider three problems connected with coinvariant subspaces of the backward shift operator in Hardy spaces $H^p$: 1) properties of truncated Toeplitz operators; 2) Carleson-type embedding theorems for the coinvariant subspaces; 3) factorizations of pseudocontinuable functions from $H^1$. These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by Donald Sarason.

Nearly invariant subspaces with applications to truncated Toeplitz operators

In this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invar…

矩阵值截断Toeplitz算子的乘积

Product of Matrix Valued Truncated Toeplitz Operators

本文研究矩阵值截断Toeplitz算子乘积仍为同类算子的条件，在补充假设下给出了乘积 $A_ΦA_Ψ$ 为矩阵值截断Toeplitz算子的充分必要条件。

Matrices de Toeplitz tronquées sur des polygones convexes. Cas du triangle

We consider the class of positive bounded and semi-continuous functions defined on the two dimensional torus If f belongs to this class, then f will be considered as the symbol of a Toeplitz operator truncated on a triangle parametrised by an integer number . We develop a geometric structure of the inverse of the Toeplitz operator and give an asymptotical development of the trace of its inverse wich brings out the geometry of the triangle. The foundation of this result consists in the possibility of f having a factorisation of type |g|^2 where the spectrum of g will be localised in a given sem…

Symmetric matrix representations of truncated Toeplitz operators on finite dimensional spaces

In this paper, we study matrix representations of truncated Toeplitz operators with respect to orthonormal bases which are invariant under a canonical conjugation map. In particular, we determine necessary and sufficient conditions for when a 3-by-3 symmetric matrix is the matrix representation of a truncated Toeplitz operator with respect to a given conjugation invariant orthonormal basis. We specialise our result to the case when the conjugation invariant orthonormal basis is a modified Clark basis. As a corollary to this specialisation, we answer a previously stated open conjecture in the n…

Norms of truncated Toeplitz operators and numerical radii of restricted shifts

This paper gives a new approach to the calculation of the numerical radius of a restricted shift operator by linking it to the norm of a truncated Toeplitz operator (TTO), which can be be calculated by various methods. Further results on the norm of a TTO are derived, and a conjecture on the existence of continuous symbols for compact TTO is resolved.

配对核与截断Toeplitz算子

Paired kernels and truncated Toeplitz operators

该文在单位圆上$L^2$与$H^2$空间框架下研究配对算子及其核（含Toeplitz核推广的解析投影核），厘清核的包含关系并应用于有限秩非对称截断Toeplitz算子的核刻画。

向量值Hardy空间上的Toeplitz算子与Hankel算子的斜交换子

The Skew Commutators of Toeplitz and Hankel operators in vector-valued Hardy Space

本文刻画向量值Hardy空间上Toeplitz与Hankel算子互为斜交换子的条件及乘积自伴的充要条件，分类了单侧移位算子及其伴随算子的斜交换子，推广了已有相关结果。

截断 Toeplitz 算子的 Szegő 极限定理

Szegő Limit Theorem for Truncated Toeplitz Operators

本文研究 Szegő 极限定理对截断 Toeplitz 算子的推广，提出有限维模型空间压缩情形下的新定理，并刻画了 Strouse-Timotin-Zarrabi 2017 年变体成立的序列条件。

具有算子值正规符号的次正规Toeplitz算子

Hyponormal Toeplitz operators with operator valued normal symbols

本文刻画了具有算子值正规符号的次正规Toeplitz算子，并探讨了该类算子何时为解析或正规。主要贡献在于给出了此类算子的结构特征。研究问题涉及算子理论中的次正规性，方法基于符号的算子值性质分析。结果对理解Toeplitz算子的分类有重要意义。

复对称矩阵的半代数维数与截断Toeplitz模型

Semialgebraic Dimension and Truncated Toeplitz Models for Complex Symmetric Matrices

研究复对称矩阵相关模型理论问题，利用半代数维数证明定理，表明\(n\geq10\)时存在不可约对称\(n\times n\)矩阵不酉等价于截断Toeplitz算子，还对相关精细表示问题给出肯定结果。

Non-commutative Clark measures for the free and abelian Toeplitz algebras

We construct a non-commutative Aleksandrov-Clark measure for any element in the operator-valued free Schur class, the closed unit ball of the free Toeplitz algebra of vector-valued full Fock space over $\mathbb{C} ^d$. Here, the free (analytic) Toeplitz algebra is the unital weak operator topology (WOT)-closed algebra generated by the component operators of the free shift, the row isometry of left creation operators. This defines a bijection between the free operator-valued Schur class and completely positive maps (non-commutative AC measures) on the operator system of the free disk algebra, t…

大向量值Fock空间上的Toeplitz算子

Toeplitz operators on large vector-valued Fock spaces

本文刻画了大向量值Fock空间上Toeplitz算子的有界性、紧性及Schatten类性质，通过引入算子值Berezin变换和平均函数，并利用Carleson测度给出完整表征。

Toeplitz算子在向量值Hardy空间上的扰动

Perturbations of Toeplitz operators on vector-valued Hardy spaces

本文研究了向量值Hardy空间中Toeplitz算子的有限秩扰动，分类其不变子空间并刻画了相关近似不变子空间的性质。

On some algebras of truncated Hankel and asymmetric truncated Hankel operators

In the last decade, a large amount of research has been concentrated on the operators living on the model space. Asymmetric truncated Toeplitz operators and asymmetric truncated Hankel operators are the natural generalization of truncated Toeplitz operators and truncated Hankel operators respectively. In this paper, we obtained the basic results concerning the product of these operators and in terms of product their connection with each other. In addition, when the inner function has a certain symmetric property, some algebraic properties of truncated Hankel operators are also discussed.

Which subnormal Toeplitz operators are either normal or analytic?

We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmos's Problem 5: Which subnormal block Toeplitz operators are either normal or analytic? We extend and prove Abrahamse's Theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two analytic functions), whose co-analytic part has a "coprime decomposition," is normal or analytic. We also prove that the coprime decomposition condition is essential. Finally, we …

Kernels of Perturbed Toeplitz Operators in vector-valued Hardy spaces

Recently, Liang and Partington \cite{YP} show that kernels of finite-rank perturbations of Toeplitz operators are nearly invariant with finite defect under the backward shift operator acting on the scalar-valued Hardy space. In this article we provide a vectorial generalization of a result of Liang and Partington. As an immediate application we identify the kernel of perturbed Toeplitz operator in terms of backward shift-invariant subspaces in various important cases by applying the recent theorem (\cite{CDP, OR}) in connection with nearly invariant subspaces of finite defect for the backward …

Abrahamse's Theorem for matrix-valued symbols and subnormal Toeplitz completions

This paper deals with subnormality of Toeplitz operators with matrix-valued symbols and, in particular, with an appropriate reformulation of Halmos's Problem 5: Which subnormal Toeplitz operators with matrix-valued symbols are either normal or analytic? In 1976, M. Abrahamse showed that if $φ\in L^\infty$ is such that $φ$ or $\overlineφ$ is of bounded type and if $T_φ$ is subnormal, then $T_φ$ is either normal or analytic. In this paper we establish a matrix-valued version of Abrahamse's Theorem and then apply this result to solve the following Toeplitz completion problem: Find the unspecified…

Products of Toeplitz Operators on a Vector Valued Bergman Space

We give a necessary and a sufficient condition for the boundedness of the Toeplitz product $T_FT_{G^*}$ on the vector valued Bergman space $L_a^2(\mathbb{C}^n)$, where $F$ and $G$ are matrix symbols with scalar valued Bergman space entries. The results generalize existing results in the scalar valued Bergman space case. We also characterize boundedness and invertibility of Toeplitz products $T_FT_{G^*}$ in terms of the Berezin transform, generalizing results found by Zheng and Stroethoff for the scalar valued Bergman space.