Stephan Ramon Garcia, William T. Ross
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.
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We give a complete description of the finite-rank 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.
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…
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 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.
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.
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。
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.
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.
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 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.
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.
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.
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.
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 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.
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.
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.
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…
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.
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 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.
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.
Matrix valued asymmetric truncated Toeplitz operator are compression of multiplication operators acting between two model spaces. These are the generalization of matrix valued truncated Toeplitz operators. In this paper we use generalized Crofoot transform to characterize symbol of matrix valued asymmetric truncated Toeplitz operator equal to zero operator.
In this paper we investigate operators unitarily equivalent to truncated Toeplitz operators. We show that this class contains certain sums of tensor products of truncated Toeplitz operators. In particular, it contains arbitrary inflations of truncated Toeplitz operators; this answers a question posed by Cima, Garcia, Ross, and Wogen.
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.
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".
We find necessary and sufficient conditions for the product of two truncated Toeplitz operators on a model space to itself be a truncated Toeplitz operator, and as a result find a characterization for the maximal algebras of bounded 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.
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…
We prove a spatial Isomorphism between two spaces of matrix valued 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.
On matrix valued (asymmetric) truncated Toeplitz operators
本文研究了矩阵值非对称截断Toeplitz算子的性质，定义了新的共轭运算并探讨其与Hankel算子的关系。
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.
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 were introduced in a seminal paper by Sarason in 2007. We show that they have an intimate connection with the Agler-Young class.
Truncated Toeplitz operators are C--symmetric with respect to the canonical conjugation given on an appropriate model space. However, by considering only one conjugation one cannot characterize truncated Toeplitz operators. It will be proved, for some classes of inner functions and the model spaces connected with them, that if an operator on a model space is C--symmetric for a certain family of conjugations in the model space, then is has to be truncated Toeplitz. A characterization of classical Toeplitz operators is also presented in terms of conjugations.
Equivalence after extension for general Toeplitz operators
本文综述延拓后等价性在Toeplitz算子相关算子中的应用，指出配对算子的关键作用，研究广义Toeplitz算子的可逆性与Fredholm性质，取得仅用有限信息即可完整描述其核的新成果。
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.
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.
Some properties of unbounded truncated Toeplitz operators
本文研究了在模型空间上由内函数u定义的闭稠密定义的无界截断Toeplitz算子，研究其与修改压缩位移算子的交换性及可逆性和自伴性性质。
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。
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。
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…
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。
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…
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。
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。
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。
Multipliers and equivalence of functions, spaces, and operators
本文通过闭子空间之间的乘子建立统一框架，研究 L^2 上各类广义 Toeplitz 算子的等价性，并推广了 Brown–Halmos 定理。
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.
Toeplitz, Hankel, de Branges and two truncated matrix moment problems
本文借助de Branges型向量值整函数再生核希尔伯特空间及对应开单位圆盘的同类空间，分别研究截断矩阵Hamburger矩与三角矩问题，用再生核计算投影以简化推导、得到简洁公式。
The property of being shift invariant and being reflexive or transitive in the case of the space of (asymmetric) truncated Toeplitz operators, and the space of (asymmetric) dual truncated operators is investigated. Most of the results obtained are new even for the symmetric case. A characterization of asymmetric dual truncated Toeplitz operators is also given.
On the minimum modulus of dual truncated Toeplitz operators
本文研究了双截断Toeplitz算子的最小模，推导了其显式公式并给出了谱界和条件，通过分析相关算子的范数，提供了精确公式和双侧估计，并通过实例验证了结果。
It is shown that the kernel of a Toeplitz operator with $2\times 2$ symbol $G$ can be described exactly in terms of any given function in a very wide class, its image under multiplication by $G$, and their left inverses, if the latter exist. As a consequence, under many circumstances the kernel of a block Toeplitz operator may be described as the product of a space of scalar complex-valued functions by a fixed column vector of functions. Such kernels are said to be of scalar type, and in this paper they are studied and described explicitly in many concrete situations. Applications are given to…
Norm attaining dual truncated Toeplitz operators
本文研究了双截断Toeplitz算子的范数达到问题，通过符号分解特征揭示其在分析和共分析分量上的范数达到条件，并给出了极值向量的系统描述。
Nearly invariant subspaces and weighted dual truncated Toeplitz operators
该数学研究针对近不变子空间上的加权对偶截断Toeplitz算子，证明其范数公式、紧性等性质，推导代数恒等式及相关校正、位移公式。
Brown-Halmos type theorems for generalized Cauchy singular integral operators and applications
本文研究了广义柯西奇异积分算子的可交换性及其应用，提出了统一方法分析算子类的代数性质，并给出了奇异积分算子准正规性和非对称双截断贝塞尔算子乘积性质的完整刻画。
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.
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…
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