Dmitry V. Yakubovich
For a continuous complex-valued function g on the real line without zeros, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of det G. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made.
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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。
Recently the authors characterized the Fredholmn properties of Toeplitz operators on weighted Fock spaces when the Laplacian of the weight function is bounded below and above. In the present work the authors extend their characterization to doubling Fock spaces with a subharmonic weight whose Laplacian is a doubling measure. The geometry induced by the Bergman metric for doubling Fock spaces is much more complicated than that of the Euclidean metric used in all the previous cases to study Fredholmness, which leads to considerably more involved calculations.
We consider Toeplitz and Hankel operators with piecewise continuous generating functions on $l^p$-spaces and the Banach algebra generated by them. The goal of this paper is to provide a transparent symbol calculus for the Fredholm property and a handy formula for the Fredholm index for operators in this algebra.
Fredholm index of Toeplitz pairs with $H^{\infty}$ symbols
本文刻画了双圆盘Hardy空间上具$H^{\infty}$符号的Toeplitz对的Fredholm性，基于双圆盘Corona问题的$L^p$解得到其指标公式，解决了该类算子对的Fredholm性刻画与指标计算问题。
Toeplitz operators on spaces $H^p(G)\ (1< p<\infty)$ associated with compact connected Abelian group $G$ with ordered dual are considered and the generalization of the classical Gohberg-Krein theorem on the Fredholm index of such operators with continuous symbols is proved. Applications to spectral theory of Toeplitz operators are given and examples of evident computation of index have been considered.
The Fredholm property of Toeplitz operators on the $p$-Fock spaces $F_α^p$ on $\mathbb{C}^n$ is studied. A general Fredholm criterion for arbitrary operators from the Toeplitz algebra $\mathcal{T}_{p,α}$ on $F_α^p$ in terms of the invertibility of limit operators is derived. This paper is based on previous work, which establishes corresponding results on the unit balls $\mathbb{B}^n$.
We develop a complete Fredholm and invertibility theory for Toeplitz+Hankel operators $T(a)+H(b)$ on the Hardy space $H^p$, $1<p<\infty$, with piecewise continuous functions $a,b$ defined on the unit circle which are subject to the condition $a(t)a(t^{-1})=b(t)b(t^{-1})$, $|t|=1$. In particular, in the case of Fredholmness, formulas for the defect numbers are established. The results are applied to several important examples.
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 concerns the analysis of an unbounded Toeplitz-like operator generated by a rational matrix function having poles on the unit circle T. It extends the analysis of such operators generated by scalar rational functions with poles on T found in [11,12,13]. A Wiener-Hopf type factorization of rational matrix functions with poles and zeroes on T is proved and then used to analyze the Fredholm properties of such Toeplitz-like operators. A formula for the index, based on the factorization, is given. Furthermore, it is shown that the determinant of the matrix function having no zeroes on T …
In this paper a definition is given for an unbounded Toeplitz-like operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed.
We will give a complete description of $\mathcal{I}$, the set of invertible quasicontinuous functions on the unit circle. After doing this, we will then classify the path-connected components of $\mathcal{I}$ and show that $\mathcal{I}$ has uncountably many path-connected components. We will then use the above classifications to characterize $\mathcal{F}$, the set of Fredholm operators of the C$^*$-algebra generated by the Toeplitz operators $T_ϕ$ with quasicontinuous symbols $ϕ$. Then we will classify the path-connected components of $\mathcal{F}$ and show that $\mathcal{F}$ also has uncounta…
A generalized winding number formula for the Witten index of a Toeplitz operator
本文把Toeplitz算子Fredholm指标的绕数公式推广至Witten指标，并给出相关迹公式。
On Toeplitz operators on compact Abelian groups and discrete Wiener--Hopf operators
本文研究了线性有序离散阿贝尔群上的维纳-霍普夫算子的Fredholm指数和谱性质，通过符号的逆傅里叶变换进行分析。
Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces $H^p$ of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in $L^p(\mathbb R)$.
A general integral formula for the spectral flow of a path of unbounded selfadjoint Fredholm operators subject to certain summability conditions is derived from the interpretation of the spectral flow as a winding number.
In this note known formulas for the product of Toeplitz operators are revisited in the context of their applications to the study of Fredholmness, boundedness of Toeplitz products, and the Berezin-Toeplitz quantization. A few open problems are also mentioned.
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。
A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Fredholm characteristics
本文研究单位圆上含极点的有理矩阵符号所生成无界类 Toeplitz 算子的核维数，在 Wiener–Hopf 型分解的附加条件下给出公式，并推广相关核刻画。
Conditions are established under which Fredholmness, Coburn's property and one- or two-sided invertibility are shared by a Toeplitz operator with matrix symbol $G$ and the Toeplitz operator with scalar symbol $\det G$. These results are based on one-sided invertibility criteria for rectangular matrices over appropriate commutative rings and related scalar corona type problems.
In this paper we study the Fredholm properties of Toeplitz operators acting on weighted Bergman spaces $A^p_ν(\mathbb{B}^n)$, where $p \in (1,\infty)$ and $\mathbb{B}^n \subset \mathbb{C}^n$ denotes the $n$-dimensional open unit ball. Let $f$ be a continuous function on the Euclidean closure of $\mathbb{B}^n$. It is well-known that then the corresponding Toeplitz operator $T_f$ is Fredholm if and only if $f$ has no zeros on the boundary $\partial\mathbb{B}^n$. As a consequence, the essential spectrum of $T_f$ is given by the boundary values of $f$. We extend this result to all operators in the…
A classical problem in operator theory has been to determine the spectrum of Toeplitz-like operators on Hilbert spaces of vector-valued holomorphic functions on the open unit ball in C^m. In this note we obtain necessary conditions for k-tuples of such operators to be Fredholm in the sense of Taylor and show they are sufficient in the case of the unit disk.
In this paper we prove an invertibility criterion for certain operators which is given as a linear algebraic combination of Toeplitz operators and Fourier multipliers acting on the Hardy space of the unit disc. Very similar to the case of Toeplitz operators we prove that such operators are invertible if and only if they are Fredholm and their Fredholm index is zero. As an application we prove that for "quasi-parabolic" composition operators the spectra and the essential spectra are equal.
A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Invertibility and Riccati equations
针对单位圆上含极点的有理矩阵符号所诱导的无界类 Toeplitz 算子，本文用状态空间方法将其可逆性归结为相关非对称离散代数 Riccati 方程的稳定解，并给出伪典范分解与逆算子公式。
We give a characterization of compact and Fredholm operators on polyanalytic Fock spaces in terms of limit operators. As an application we obtain a generalization of the Bauer-Isralowitz theorem using a matrix valued Berezin type transform. We then apply this theorem to Toeplitz and Hankel operators to obtain necessary and sufficient conditions for compactness. As it turns out, whether or not a Toeplitz or Hankel operator is compact does not depend on the polyanalytic order. For Hankel operators this even holds on the true polyanalytic Fock spaces.
It is well known that a Toeplitz operator is invertible if and only if its symbols admits a canonical Wiener-Hopf factorization, where the factors satisfy certain conditions. A similar result holds also for singular integral operators. More general, the dimension of the kernel and cokernel of Toeplitz or singular integral operators which are Fredholm can be expressed in terms of the partial indices of an associated Wiener-Hopf factorization problem. In this paper we establish corresponding results for Toeplitz + Hankel operators and singular integral operators with flip under the assumption th…
A Borodin-Okounkov-Geronimo-Case identity for tilted Toeplitz minors
本文证明了倾斜Toeplitz子式的Fredholm行列式恒等式，将BOGC恒等式推广到Hardy空间的斜分割，并应用于周期TASEP中KPZ不动点的识别。
The Helton-Howe measure of almost normal Toeplitz operators
本文研究了近正交 Toeplitz 运算符的 Helton-Howe 测度，通过符号的调和扩展确定其测度，并推广了 Toeplitz 运算符的 Fredholm 指数 winding 数公式。
Toeplitz operators on non-reflexive Fock spaces
本文将自反 Fock 空间上 Toeplitz 算子的紧性、Fredholm 性质及 Toeplitz 代数等结果推广至非自反情形，并改进了相关对应理论。
For operators belonging either to a class of global bisingular pseudodifferential operators on $R^m \times R^n$ or to a class of bisingular pseudodifferential operators on a product $M \times N$ of two closed smooth manifolds, we show the equivalence of their ellipticity (defined by the invertibility of certain associated homogeneous principal symbols) and their Fredholm mapping property in associated scales of Sobolev spaces. We also prove the spectral invariance of these operator classes and then extend these results to the even larger classes of Toeplitz type operators.
For bounded Lebesgue measurable functions $f,g,ϕ$ and $ψ$ on the unit circle, $P_{+}fP_{+}+P_{-}gP_{+} +P_{+}ϕP_{-}+P_{-}ψP_{-}$ is called a generalized singular integral operator (GSIO) on $L^{2}(\mathbb{T})$, where $P_{+}$ is the Riesz projection, $P_{-}=I-P_{+}.$ In this paper, we relate GSIOs to a number of operators, including Cauchy singular integral operator, (dual) truncated Toeplitz operator, Foguel-Hankel operator, multiplication operator, Toeplitz plus Hankel operator etc. We establish the short exact sequences associated of the $C^{*}-$algebras generated by GSIOs with bounded or qu…
Suppose that $ϕ$ and $ψ$ are smooth complex-valued functions on the circle that are invertible, have winding number zero with respect to the origin, and have meromorphic extensions to an open neighborhood of the closed unit disk. Let $T_ϕ$ and $T_ψ$ denote the Toeplitz operators with symbols $ϕ$ and $ψ$ respectively. We give an explicit formula for the determinant of $T_ϕT_ψT_ϕ^{-1} T_ψ^{-1}$ in terms of the products of the tame symbols of $ϕ$ and $ψ$ on the open unit disk.
We introduce a new topology, weaker than the gap topology, on the space of selfadjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of selfadjoint Breuer-Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips' analytical definition and derive integral formulas for the spectral flow for certain paths of unbounded operators with common domain, generalizing those of Carey--Phillips. Furthermore we prove the homotopy invariance of the real-valued index. As an example w…
Let $M(ϕ)=T(ϕ)+H(ϕ)$ be the Toeplitz plus Hankel operator acting on $H^p(\T)$ with generating function $ϕ\in L^\iy(\T)$. In a previous paper we proved that $M(ϕ)$ is invertible if and only if $ϕ$ admits a factorization $ϕ(t)=ϕ_{-}(t)ϕ_{0}(t)$ such that $ϕ_{-}$ and $ϕ_{0}$ and their inverses belong to certain function spaces and such that a further condition formulated in terms of $ϕ_{-}$ and $ϕ_{0}$ is satisfied. In this paper we prove that this additional condition is equivalent to the Hunt-Muckenhoupt-Wheeden condition (or, $A_{p}$-condition) for a certain function $σ$ defined on $[-1,1]$, w…
Unbounded (and bounded) Toeplitz operators (TO) with rational symbols are analysed in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are determined. In particular, in the symmetric case, i.e., for a real rational symbol the deficiency spaces and indices are explicitly available. The concluding section gives a brief overview on the research on unbounded TO in order to locate the present contribution. Regarding properties of unbounded TO in general, it fu…
Spectral properties of Toeplitz operators with harmonic function symbols on the Bergman space
本文研究了贝拉曼空间上具有调和函数符号的托埃利茨算子的谱性质，揭示了其谱结构和Fredholm指数特性。
This paper is a follow-up to a recent article about the essential spectrum of Toeplitz operators acting on the Bergman space over the unit ball. As mentioned in the said article, some of the arguments can be carried over to the case of bounded symmetric domains and some cannot. The aim of this paper is to close the gaps to obtain comparable results for general bounded symmetric domains. In particular, we show that a Toeplitz operator on the Bergman space $A^p_ν$ is Fredholm if and only if all of its limit operators are invertible. Even more generally, we show that this is in fact true for all …
Asymptotics of block Toeplitz determinants with piecewise continuous symbols
本研究针对带有限跳跃的矩阵值分段连续符号，在温和条件下确定块Toeplitz行列式的渐近性，提出新的Toeplitz行列式局部化定理与Fredholm指标计算方法，结果可应用于量子自旋链模型纠缠熵研究。
This is an expository article. It discusses an approach to hypoelliptic Fredholm index theory based on noncommutative methods (groupoids, C*-algebras, K-theory). The paper starts with an explicit index theorem for scalar second order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-brow index formula is expressed in terms of winding numbers. We then proceed to show how this theorem is a special case of a much more general index theorem for subelliptic operators on contact manifolds. Finally we discuss the noncommutative topology that is employed in the proof o…
On the dynamics of Toeplitz operators over Bergman spaces
本文研究 Bergman 空间上 Toeplitz 算子的超循环性，通过直接求解预解方程和绕数论证，在三对角情形下给出精确充要条件，消除了先前限制。
Coburn's Lemma and the Finite Section Method for Random Jacobi Operators
本文研究了有限和无限三对角随机矩阵的谱和伪谱，证明了在伪ergodic条件下，有限和无限三对角随机矩阵的谱具有特定性质，并展示了有限截断方法在求解方程和估计谱时的有效性。
Wiener-Hopf indices of unimodular functions on the imaginary axis
本文研究虚轴上单模有理矩阵函数的 Wiener-Hopf 指标，利用 Douglas-Shapiro-Shields 分解中因子实现的矩阵给出指标公式，并通过算子论直接方法与 Cayley 变换两种途径建立结果。
Let $Φ$ be a continuous $n\times n$ matrix-valued function on the unit circle $\T$ such that the $(k-1)$th singular value of the Hankel operator with symbol $Φ$ is greater than the $k$th singular value. In this case, it is well-known that $Φ$ has a unique superoptimal meromorphic approximant $Q$ in $H^{\infty}_{(k)}$; that is, $Q$ has at most $k$ poles in the unit disc $\mathbb{D}$ (i.e. the McMillan degree of $Q$ in $\mathbb{D}$ is at most $k$) and $Q$ minimizes the essential suprema of singular values $s_{j}((Φ-Q)(ζ))$, $j\geq0$, with respect to the lexicographic ordering. For each $j\geq 0$…
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