Maria T. Nowak, Paweł Sobolewski, Andrzej Sołtysiak, Magdalena Wołoszkiewicz-Cyll
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.
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We review some classical and more recent results concerning kernels of Toeplitz operators and their relations with model spaces, which are themselves Toeplitz kernels of a special kind. We highlight the fundamental role played by the existence of maximal vectors for every nontrivial Toeplitz kernel.
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 consider kernels of unbounded Toeplitz operators in $H^p(\mathbb C^+)$ in terms of a factorization of their symbols. We study the existence of a minimal Toeplitz kernel containing a given function in $H^p(\mathbb C^+)$, we describe the kernels of Toeplitz operators whose symbol possesses a certain factorization involving two different Hardy spaces and we establish relations between the kernels of two operators whose symbols differ by a factor which corresponds, in the unit circle, to a non-integer power of $z$. We apply the results to describe the kernels of Toeplitz operators with non-vani…
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。
It is well known that the kernel of a Toeplitz operator is nearly invariant under the backward shift $S^*$. This paper shows that kernels of finite-rank perturbations of Toeplitz operators are nearly $S^*$-invariant with finite defect. This enables us to apply a recent theorem by Chalendar--Gallardo--Partington to represent the kernel in terms of backward shift-invariant subspaces, which we identify in several important cases.
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 characterize operator-theoretic properties (boundedness, compactness, and Schatten class membership) of Toeplitz operators with positive measure symbols on Bergman spaces of holomorphic hermitian line bundles over Kähler Cartan-Hadamard manifolds in terms of geometric or operator-theoretic properties of measures.
This paper considers model spaces in an $H_p$ setting. The existence of unbounded functions and the characterisation of maximal functions in a model space are studied, and decomposition results for Toeplitz kernels, in terms of model spaces, are established.
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…
Kernel-Summability Methods and the Silverman-Toeplitz Theorem
本文引入 Banach 空间中的核可和方法，建立其 Silverman–Toeplitz 型定理与标量到向量值情形的包含关系推广，并用于研究全纯函数 Taylor 级数的可和性。
In this paper we study the kernels of Toeplitz operators on both the scalar and the vector-valued Hardy space for $ 1 < p < \infty $. We show existence of a minimal kernel of any element of the vector-valued Hardy space and we determine a symbol for the corresponding Toeplitz operator. In the scalar case we give an explicit description of a maximal function for a given Toeplitz kernel. In the vectorial case we show not all Toeplitz kernels have a maximal function and in the case of $p=2$ we find the exact conditions for when a Toeplitz kernel has a maximal function. For both the scalar and vec…
For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman kernel. Finally, we also exhibit a holomorphic continuation of the kernels with respect to the Sobolev parameter to the entire complex plane. Our main tool are the generalized Toeplitz operators of Bout…
Maximal functions, conjugations and multipliers between Toeplitz kernels
本文研究了Toeplitz核通过Riemann-Hilbert问题、最大函数和乘子的多种定义方式及其关系，重点探讨了符号因子分解在构造乘子中的作用以及最大函数与共轭运算的联系。
The recent theorem by D. Luecking that finite rank Toeplitz-Bergman operators must be generated by a measure consisting of finitely many point masses is carried over to the many-dimensional case.
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…
We give a complete description of the finite-rank truncated Toeplitz operators.
Construction of analytic functions, which determine bounded Toeplitz operators
Paired kernels and their applications
本文在Hardy空间$H^2$中研究配对算子的核及其解析投影，获得近不变性、表示和包含关系，并证明最小Toeplitz核的存在性，进而描述有限秩非对称截断Toeplitz算子的核。
We study Toeplitz operators on the Bargmann space, with Toeplitz symbols that are exponentials of complex quadratic forms, from the point of view of Fourier integral operators in the complex domain. Sufficient conditions are established for the composition of two such operators to be a Toeplitz operator.
We construct explicit examples of non-trivial nilpotent Toeplitz operators on Bergman spaces of certain Reinhardt domains in $\mathbb{C}^2$.
We study Toeplitz operators on the Bargmann space, with Toeplitz symbols that are exponentials of inhomogeneous quadratic polynomials. It is shown that the boundedness of such operators is implied by the boundedness of the corresponding Weyl symbols.
In this paper we discuss the multipliers between range spaces of co-analytic Toeplitz operators.
We study algebraic properties of Toeplitz operators on Bergman spaces of polyanalytic functions on the unit disk. We obtain results on finite-rank commutators and semi-commutators of Toeplitz operators with harmonic symbols. We also raise and discuss some open questions.
Schatten-Herz class Toeplitz operators on weighted Bergman spaces induced by doubling weights are investigated in this paper.
Maximal functions and multipliers for subkernels of Toeplitz operators
本文研究了Toeplitz算子子核中极值函数与乘子的关系，通过模型空间表示方法揭示了重要Toeplitz核的结构特性。
Given a polyanalytic function, we show that the corresponding Toeplitz operator on the Bergman space of the unit disc can be expressed as a quotient of certain differential operators with holomorphic coefficients. This enables us to obtain several operator theoretic results including a criterion for invertibility of a Toeplitz operator.
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.
We introduce a new class of conjugations and characterize complex symmetric Toeplitz operators on the Hardy space with respect to those conjugations. Also, we prove that complex symmetricity and \uet~ property are the same for a certain class of Toeplitz operators. We also discuss the analytic symmetricity for Toeplitz operators. Our results extend several known results by providing unified ways of treating them.
We discuss resent developments in the problem of description of finite rank Toeplitz operators in different Bergman spaces and give some applications in analysis and mathematical physics
The boundedness and compactness of Toeplitz operator from $A_ω^p$ to $A_ω^q$ with doubling weights $ω$ are studied in this paper. The characterizations of Schatten class Toeplitz operators and Volterra operators on $A_ω^2$ are also investigated.
In this article we characterize the boundedness and compactness of a Toeplitz-type operator on weighted Bergman spaces satisfying the so-called Bekolle-Bonami condition in terms of the Berezin transform.
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 …
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。
Given a CR manifold with non-degenerate Levi form, we show that the operators of the functional calculus for Toeplitz operators are complex Fourier integral operators of Szegő type. As an application, we establish semi-classical spectral dimensions for Toeplitz operators. We then consider a CR manifold with a compact Lie group action $G$ and we establish quantization commutes with reduction for Toeplitz operators. Moreover, we also compute semi-classical spectral dimensions for $G$-invariant Toeplitz operators.
In this note we describe the commutant of the multiplication operator by a monomial in the Toeplitz algebra of a complete strongly pseudoconvex Reinhardt domain.
Multiplier Between Generalized Toeplitz Kernels
本文发展了广义 Toeplitz 核之间乘子的结构分类，扩展了 Fricain 和 Rupam 的工作，建立了乘子空间与 Carleson 型嵌入的新等价关系，并将其与 Beurling Malliavin 密度、Pólya 序列及整函数谱理论相联系。
Toeplitz plus Hankel operators $T(a)+H(b)$, $a,b\in L^\infty$ acting on the classical Hardy spaces $H^p, 1<p<\infty$, are studied. If the generating functions $a$ and $b$ satisfy the so-called matching condition $a(t) a(1/t)=b(t) b(1/t)$, an effective description of the structure of the kernel and cokernel of the corresponding operator is given. The results depend on the behaviour of two auxiliary scalar Toeplitz operators, and if the generating functions $a$ and $b$ are piecewise continuous, more detailed results are obtained.
We study Toeplitz operators on the Bargmann space, with Toeplitz symbols given by exponentials of complex quadratic forms. We show that the boundedness of the corresponding Weyl symbols is necessary for the boundedness of the operators, thereby completing the proof of the Berger-Coburn conjecture in this case. We also show that the compactness of such Toeplitz operators is equivalent to the vanishing of their Weyl symbols at infinity.
The recent theorem by D.Luecking about finite rank Bergman-Toeplitz operators is extended to weights being distributions with compact support and to the spaces of harmonic functions.
We study the intertwining relations between analytic Toeplitz operators induced on the Hardy space H^2 by analytic functions bounded on the open unit disc. Our work centers on the connection between intertwining between the Toeplitz operators the image containment between their symbols, as well as on the nature of the intertwining operator. We use our results to study the "extended eigenvalues" of analytic Toeplitz operators, i.e., the special case where the operator is intertwined with a scalar multiple of itself.
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 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.
The paper is concerned with a sequence of constants which appear in several problems. These problems include the minimal eigenvalue of certain positive definite Toeplitz matrices, the minimal eigenvalue of some higher-order ordinary differential operators, the norm of the Green kernels of these operators, the best constant in a Wirtinger-Sobolev inequality, and the conditioning of a special least squares problem. The main result of the paper gives the asymptotics of this sequence.
We compute the Dixmier trace of pseudo-Toeplitz operators on the Fock space. As an application we find a formula for the Dixmier trace of the product of commutators of Toeplitz operators on the Hardy and weighted Bergman spaces on the unit ball of \mathbb C^d. This generalizes an earlier work of Helton-Howe for the usual trace of the anti-symmetrization of Toeplitz operators.
Bergman Projections, Kernel $p$-Norm Estimates, and Toeplitz Operators with Békollé and Bonami weights
本文针对加权贝科勒 - 博纳米伯格曼空间间的托普利兹算子，通过建立核p-范数估计来刻画其有界和紧性，借助广义贝雷津变换等描述性质，引入权重条件确保加权伯格曼投影有界，将相关结果从希尔伯特空间推广到巴拿赫空间。
We study the boundedness and compactness of the generalized Volterra integral operator on weighted Bergman spaces with doubling weights on the unit disk. A generalized Toeplitz operator is defined and the boundedness, compactness and Schatten class of this operator are investigated on the Hilbert weighted Bergman space. As an application, Schatten class membership of generalized Volterra integral operators are also characterized. Finally, we also get the characterizations of Schatten class membership of generalized Toeplitz operator and generalized Volterra integral operators on the Hardy spac…
Toeplitz Operators and Berezin-type Operators on Different Bergman Spaces
本文研究管状域上不同加权Bergman空间之间Toeplitz算子和Berezin型算子的有界性与紧性，建立其与Carleson测度的联系并给出刻画。
A counterexample of the Fredholm of Toeplitz operator
本文通过构造Hardy空间$H^1$上Toeplitz算子的一个反例，证明其不是Fredholm算子，从而否定了Virtanen在2006年提出的猜想。
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.
We study positive Toeplitz operators on the Bergman spaces having the fast decreasing weight functions in a certain class. We give the characterizations for the boundedness and compactness of Toeplitz operators in terms of their Berezin transforms and averaging functions. In particular, we show the equivalent conditions in order that Toeplitz operators on $A^2(ω)$ belong to the Schatten class $S_p$, $0<p<\infty$.
A full description of the membership in the Schatten ideal $S_ p(A^2_ω)$ for $0<p<\infty$ of the Toeplitz operator acting on large weighted Bergman spaces is obtained.
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.
In this paper, we characterize operator-theoretic properties (boundedness, compactness, and Schatten class membership) of Toeplitz operators with positive measure symbols on weighted Fock-Sobolev spaces of fractional order.
Using the approach based on sesquilinear forms, we introduce Toeplitz operator in the analytic Bergman space on the upper half-plane with strongly singular symbols, derivatives of measures. Conditions for boundedness and compactness of such operators are found. A procedure of reduction of Toeplitz operators in Bergman spaces of polyanalytic functions to operators with singular symbols in the analytic Bergman space by means of the creation-annihilation structure is elaborated, which leads to the description of the properties of the former 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…
Nearly invariant subspaces and kernels of Toeplitz operators on the Hardy space over the bidisk
本文研究双圆盘Hardy空间中Toeplitz算子的近似不变子空间与核，通过推广向量值Hardy空间结果、定义双圆盘及交换等距元组的近似不变子空间，刻画并证明了Toeplitz算子核的近似不变性。
On the Localization of the Bergman Kernel and applications to Toeplitz theory
研究Bergman核在大线丛和Radon测度下的局部化现象，确认Zelditch猜想，并展示Toeplitz算子在组合下构成代数，其谱在Bernstein-Markov测度下均匀分布。
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.
Generalized Toeplitz plus Hankel operators $T(a)+H_α(b)$ generated by functions $a,b$ and a linear fractional Carleman shift $α$ changing the orientation of the unit circle $\mathbb{T}$ are considered on the Hardy spaces $H^p(\mathbb{T})$, $1<p<\infty$. If the functions $a,b\in L^\infty(\mathbb{T})$ and satisfy the condition $$ a(t) a(α(t))=b(t) b(α(t)),\quad t\in \mathbb{T}, $$ the defect numbers of the operators $T(a)+H_α(b)$ are established and an explicit description of the structure of the kernels and cokernels of the operators mentioned is given.
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