Joanna Jurasik, Bartosz Łanucha
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.
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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.
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.
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.
Commutation relations for truncated Toeplitz operators
截断Toeplitz算子，这是压缩的乘法算子模型的子空间的Hardy空间$H^2$，我们得到的交换关系的标准。这些结果与Toeplitz矩阵的情形类似，推广了Sedlock代数的理论。
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.
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".
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…
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.
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.
Bounded symbols and reproducing kernel thesis for truncated Toeplitz operators
Toeplitz算子到$H^2$的余不变子空间的可积称为截断Toeplitz算子。我们研究了与这些算子相关的两个问题。第一，提出了Sarason，是是否有界的运营商意味着存在一个有界的符号;第二是再生核论文。我们表明，在一般情况下，第一个问题的答案是否定的，我们展示了一些类的空间，这两个问题的答案是肯定的。
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.
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…
Truncated Toeplitz operators of finite rank
我们给出了有限秩截断Toeplitz算子的完整描述。
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.
Unitary equivalence to truncated Toeplitz operators
本文研究了与截断Toeplitz算子酉等价的算子。证明了这类算子包含截断Toeplitz算子的某些张量积和。特别是，它包含任意通货膨胀截断Toeplitz运营商，这回答了一个问题所提出的西玛，加西亚，罗斯和沃根。
Algebras of truncated Toeplitz operators
本文给出了模型空间上两个截断Toeplitz算子的乘积为截断Toeplitz算子的充要条件，并由此给出了有界截断Toeplitz算子的极大代数的一个刻划.
We prove a spatial Isomorphism between two spaces of matrix valued truncated Toeplitz operators.
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。
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.
The Norm of a Truncated Toeplitz Operator
我们证明了截断Toeplitz算子范数的几个下界，并得到了模型空间中函数的$H^2$和$H^{\infty}$范数之间的一个奇怪的关系。
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。
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.
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。
$C^*$-algebras generated by truncated Toeplitz operators
我们得到一个类似的Coburn的描述的Toeplitz代数的设置截断Toeplitz运营商。作为副产品，我们提供了几个复杂的对称算子的例子，这是不酉等价于截断的Toeplitz算子具有连续的符号。
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。
Spatial isomorphisms of algebras of truncated Toeplitz operators
研究了截断Toeplitz算子中的两个极大交换代数空间同构的条件。这是建立在最近的工作N。Sedlock，他获得了截断Toeplitz算子的极大代数的完整描述。
Truncated Toeplitz Operators: Spatial Isomorphism, Unitary Equivalence, and Similarity
截断Toeplitz算子是Toeplitz算子$T_ϕ:H^2\to H^2$到模型空间$\K_Θ := H^2 \ominus ΘH^2$的压缩$A_ϕ:\K_Θ \to \K_Θ$。对于$Θ$内部，令$\T_Θ$表示$\K_Θ$上所有有界截断Toeplitz算子的集合。我们的主要结果是关于内函数$Θ_1$和$Θ_2$的一个充分必要条件，它保证了$\mathcal{T}_{Θ_1}$和$\mathcal{T}_{Θ_2}$是空间同构的（即，$U\T_{Θ_1} = \T_{Θ_2}U$，对于某些单一的$U:\K_{Θ_1} \to \K_{Θ_2}$）。我们还研究了截断Toeplitz算子的酉等价算子，证明了有限维Hilbert空间上的每一个算子都类似于截断Toeplitz算子。
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.
Schatten-class Truncated Toeplitz Operators
我们研究截断Toeplitz算子属于Schatten理想。我们完全刻画了这样的运营商时，他们有一个解析符号或属于理想的希尔伯特-施密特运营商。我们还研究了模型空间产生的Blaschke产品与薄序列，模型空间产生的某些类型的奇异内函数，和运营商与一类非常光滑的符号。
Symbols of truncated Toeplitz operators
本文研究了Hardy空间$H^p$中后移位算子的余不变子空间的三个问题：1）截断Toeplitz算子的性质; 2）余不变子空间的Carleson型嵌入定理; 3）$H^1$中伪连续函数的因子分解.事实证明，这些问题是密切相关的，甚至在某种意义上是等同的。新的方法的基础上的因式分解，使我们能够回答一些具有挑战性的问题截断Toeplitz运营商所提出的唐纳德Sarason。
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 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。
Asymmetric truncated Toeplitz operators and Toeplitz operators with matrix symbol
在半平面的Hardy空间$H^p$中研究了截断Toeplitz算子及其非对称形式，其中$H^p$是$1<p<\infty$的半平面.它表明，他们是等价的扩展后，$2 \times 2$矩阵Toeplitz运营商，这使得人们可以推断出有关其可逆性的信息。移位模型空间的背景下，不变子空间，允许一个推导出新的Beurling-Lax定理。
Unitary equivalence to a truncated Toeplitz operator: analytic symbols
与$H^2$上的Toeplitz算子不同，截断Toeplitz算子不具有自然的矩阵特征。因此，这些运营商是很难研究数值。本文给出了特征值不同的矩阵酉等价于具有解析符号的截断Toeplitz算子的判据。这个测试是建设性的，我们用几个例子来说明它。作为一个副产品，我们还证明了每个复对称算子的希尔伯特空间的维数$\leq 3$是酉等价的截断Toeplitz算子的直和。
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…
Recent progress on truncated Toeplitz operators
本文是对截断Toeplitz算子理论的一个综述。我们首先简要介绍了这个问题，然后强调了自2007年Sarason的开创性论文以来该领域的许多最新发展。
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。
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。
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.
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。
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。
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。
Product of rectangular Toeplitz (Hankel) matrices
研究非对称Toeplitz矩阵乘积的性质，给出乘积仍为Toeplitz的充要条件，并探讨等距性相关结果。
We relate dual-band general Toeplitz operators to block truncated Toeplitz operators and, via equivalence after extension, with Toeplitz operators with $4 \times 4$ matrix symbols. We discuss their norm, their kernel, Fredhomlness, invertibility and spectral properties in various situations, focusing on the spectral properties of the dual-band shift, which turns out to be considerably complex, leading to new and nontrivial connections with the boundary behaviour of the associated inner function.
Following Beurling's theorem the natural compressions of the multiplication operator in the classical $L^2$ space are compressions to model spaces and to their orthogonal complements. Two possibly different model spaces are considered hence asymmetric truncated Toeplitz and asymmetric dual truncated Toeplitz operators are investigated. The main purpose of the paper is to characterize operators which intertwine compressions of the unilateral shift.
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。
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。
In this paper we prove that if the polar decomposition of a symbol $f$ is truncated above, i.e., $f(re^{iθ} )=\sum_{k=-\infty}^Ne^{ikθ} f_k (r)$ where the $f_k$'s are radial functions, and if the associated Toeplitz operator $T_f$ commutes with $T_{z^2+\bar{z}^2}$, then $T_f=Q(T_{z^2+\bar{z}^2})$ where $Q$ is a polynomial of degree at most $1$. This gives a partial answer to an open problem by S. Axler, Z. Cuckovic and N. V. Rao [2, p. 1953].
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