In this paper we study the Toeplitz algebra, which is generated by Toeplitz operators with bounded symbols on the Fock space $F^p_α$. We show that the Toeplitz algebra coincides with each of the algebras generated by band-dominated, sufficiently localized and weakly localized operators, respectively. Moreover, we determine its essential commutant and its essential bicommutant. For $p = 2$ these results were obtained recently by Xia. However, Xia's ideas are mostly connected to Hilbert space theory and methods which are not applicable for $p \neq 2$. Instead, we use a recent result of Fulsche to generalize Xia's theorems.
One of the major goals in the theory of Toeplitz operators on the Bergman space over the unit disk $\mathbb{D}$ in the complex place $\mathbb{C}$ is to completely describe the commutant of a given Toeplitz operator, that is, the set of all Toeplitz operators that commute with it. Here we shall study the commutants of a certain class of quasihomogeneous Toeplitz operators.
We make a progress towards describing the commutants of Toeplitz operators with harmonic symbols on the Bergman space over the unit disk. Our work greatly generalizes several partial results in the field.
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.
In this note we show that if two Toeplitz operators on a Bergman space commute and the symbol of one of them is analytic and nonconstant, then the other one is also analytic.
We make a progress towards describing the semi-commutants of Toeplitz operators on Fock-Sobolev spaces of nonnegative orders. We generalize the results in \cite{Bauer1,Qin}. For the certain symbol spaces, we obtain two Toeplitz operators can semi-commute only in the trivial cases, which is different from what is known for the classical Fock spaces. As an application, we consider the conjecture which was shown to be false for Fock space in \cite{MA}. The main results of this paper say that there is the fundamental difference between the geometries of Fock and Fock-Sobolev space.
In the setting of the Fock space over the complex plane, Bauer and Lee have recently characterized commutants of Toeplitz operators with radial symbols, under the assumption that symbols have at most polynomial growth at infinity. Their characterization states: If one of the symbols of two commuting Toeplitz operators is nonconstant and radial, then the other must be also radial. We extend this result to the Fock-Sobolev spaces.
One of the major questions in the theory of Toeplitz operators on the Bergman space over the unit disk $\mathbb D$ in the complex plane $\mathbb C$ is a complete description of the commutant of a given Toeplitz operator, that is the set of all Toeplitz operators that commute with it. In \cite{l}, the first author obtained a complete description of the commutant of Toeplitz operator $T$ with any quasihomogeneous symbol $ϕ(r)e^{ipθ}, p>0$ in case it has a Toeplitz p-th root $S$ with symbol $ψ(r)e^{iθ}$, namely, commutant of $T$ is the closure of the linear space generated by powers $S^n$ which a…
For $α>-1$, let $A^2_α$ be the corresponding weighted Bergman space of the unit ball in $\mathbb{C}^n$. For a bounded measurable function $f$, let $T_f$ be the Toeplitz operator with symbol $f$ on $A^2_α$. This paper describes all the functions $f$ for which $T_f$ commutes with a given $T_g$, where $g(z)=z_{1}^{L_1}... z_{n}^{L_n}$ for strictly positive integers $L_1,..., L_n$, or $g(z)=|z_1|^{s_1}... |z_n|^{s_n}h(|z|)$ for non-negative real numbers $s_1,..., s_n$ and a bounded measurable function $h$ on $[0,1)$.
We prove an approximation theorem on a class of domains in $\mathbb{C}^n$ on which the $\overline{\partial}$-problem is solvable in $L^{\infty}$. Furthermore, as a corollary, we obtain a version of the Axler-Čučković-Rao Theorem in higher dimensions.
When is the collection of $\mathsf S$-Toeplitz operators with respect to a tuple of commuting bounded operators $\mathsf S= (S_1, S_2, \ldots , S_{d-1}, P)$, which has the symmetrized polydisc as a spectral set, non-trivial? The answer is in terms of powers of $P$ as well as in terms of a unitary extension. En route, Brown-Halmos relations are investigated. A commutant lifting theorem is established. Finally, we establish a general result connecting the $C^*$-algebra generated by the commutant of $\mathsf S$ and the commutant of its unitary extension $\mathsf R$.
Consider a bounded strongly pseudo-convex domain $Ω$ with a smooth boundary in $\mathbb{C}^n$. Let $\mathcal{T}$ be the Toeplitz algebra on the Bergman space $L^2_a(Ω)$. That is, $\mathcal{T}$ is the $C^\ast $-algebra generated by the Toeplitz operators $\{T_f : f \in L^\infty (Ω)\}$. Extending previous work in the special case of the unit ball, we show that on any such $Ω$, $\mathcal{T}$ and $\{T_f : f \in {\text{VO}}_{\text{bdd}}\} + \mathcal{K}$ are essential commutants of each other. On a general $Ω$ considered in this paper, the proofs require many new ideas and techniques. These same tec…
Let $α:G \curvearrowright M$ be a spatial action of countable abelian group on a "spatial" von Neumann algebra $M$ and $S$ be its unital subsemigroup with $G=S^{-1}S$. We explicitly compute the essential commutant and the essential fixed-points, modulo the Schatten $p$-class or the compact operators, of the w$^*$-semicrossed product of $M$ by $S$ when $M'$ contains no non-zero compact operators. We also prove a weaker result when $M$ is a von Neumann algebra on a finite dimensional Hilbert space and $(G,S)=(\mathbb{Z},\mathbb{Z}_{+})$, which extends a famous result due to Davidson (1977) for t…
We define a new notion of entropy for operators on Fock spaces and positive definite multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (multi-Toeplitz, multi-analytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive definite multi-Toeplitz kernels on free semigroups (resp.multi-Toeplitz operators) and consequences concerning the extreme points of the unit ball of the noncommutative analytic Toeplitz algebra F_n^\infty. We obtain several geometric characterizations of the multivar…
We characterize the commutant of the analytic Toeplitz operators modulo operators of Schatten-p-class on suitable multivariable domains. We show that a result of J. Xia on compact perturbations of Toeplitz operators on the unit disc remains true on strictly pseudoconvex and symmetric domains in C^n if the ideal of compact operators is replaced by the Schatten-p-classes with p strictly less than infinity.
A family $\{T_j\}_{j\in J}$ of commuting Hilbert space operators is said to be a spherical isometry if $\sum_{j\in J}T^*_jT_j=1$ in the weak operator topology. We show that every commuting family $\Cal F$ of spherical isometries has a commuting normal extension $\hat{\Cal F}$. Moreover, if $\hat{\Cal F}$ is minimal, then there exists a natural short exact sequence $0\to\Cal C\to C^*(\Cal F)\to C^*(\hat{\Cal F})\to 0$ with a completely isometric cross-section, where $\Cal C$ is the commutator ideal in $C^*(\Cal F)$. We also show that the space of Toeplitz operators associated to $\Cal F$ is com…
We study two problems involving algebraic properties of Toeplitz operators on generalized Fock spaces on $\mathbb{C}^d$ with weights of the form $\left|z\right|^{2s} e^{-\left|z\right|^{2m}}$, $m\geq 1,\ s\geq 0$. We determine the commutant of a given Toeplitz operator with a radial symbol which satisfies certain growth conditions. We also discuss the equation $T_fT_g=0$, when $f$ or $g$ is radial.
We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ induced by finite Blachke products $B$ acting on weighted Bergman spaces which, as a particular instance, yields the case $B(z)=z^n$ on the Bergman space solved recently by by Abkar, Cao and Zhu. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces $H^p$ for $1<p<\infty$. Finally, we apply this approach to study the reducing subspaces of $T_{B}$ in weighted Bergman spaces.
We extend to the case of a threshold ideal our result with J. Bourgain about the essential centre of the commutant mod a diagonalization ideal for a n-tuple of commuting Hermitian operators . We also compute the $K_0$-group of the commutant mod trace-class of a unitary operator with spectrum equal to its essential spectrum. We present the problem of computing the $K_1$-group for a commutant mod trace-class in its simplest case.
There are three main results in this paper. First, we find an easily computable and simple condition which is necessary and sufficient for a commuting tuple of contractions to possess a non-zero Toeplitz operator. This condition is just that the adjoint of the product of the contractions is not pure. On one hand this brings out the importance of the product of the contractions and on the other hand, the non-pureness turns out to be equivalent to the existence of a pseudo-extension to a tuple of commuting unitaries. The second main result is a commutant pseudo-extension theorem obtained by stud…
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…
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 symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then stu…
We study Toeplitz operators with uniformly continuous symbols on generalized harmonic Bergman spaces of the unit ball in $\mathbb{R}^n$. We describe their essential spectra and establish a short exact sequence associated with the $C^{*}$-algebra generated by these operators.
We give estimates for the essential norms of a positive Toeplitz operator on the Bergman space of a minimal bounded homogeneous domain in terms of the Berezin transform or the averaging function of the symbol. Using these estimates, we also give necessary and sufficient conditions for the positive Toeplitz operators to be compact.
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 …
In the recent paper by Mark C. Ho (2014) the notion of a $λ$-Toeplitz operator on the Hardy space $H^2(\mathbb{T})$ over the one-dimensional torus $\mathbb{T}$ was introduced and it was shown (under the supplementary condition) that for $λ\in \mathbb{T}$ the essential spectrum of such an operator is invariant with respect to the rotation $z\mapsto λz$; if in addition $λ$ is not of finite order the essential spectrum is circular. In this paper, we generalize these results to the case when $\mathbb{T}$ is replaced by an arbitrary compact Abelian group whose dual is totally ordered.
This work is a generalization of the results in [Gul] to bi-disc case. As in [Gul], quasi-parabolic composition operators on the Hilbert-Hardy space of the bi-disc are written as a linear combination of Toeplitz operators and Fourier multipliers. The C*-algebra generated by Toeplitz operators and Fourier multipliers on the Hilbert-Hardy space of the bi-disc is written as the tensor product of the similar C*-algebra in one variable with itself. As a result we find a nontrivial set lying inside the essential spectra of quasi-parabolic composition operators.
In this paper, we show that on the weighted Bergman space of the unit disk the essential norm of a noncompact Hankel operator equals its distance to the set of compact Hankel operators and is realized by infinitely many compact Hankel operators, which is analogous to the theorem of Axler, Berg, Jewell and Shields on the Hardy space in Axler et al. (1979); moreover, the distance is realized by infinitely many compact Hankel operators with symbols continuous on the closure of the unit disk and vanishing on the unit circle.
It is well known that the essential norm of a Toeplitz operator on the Hardy space $H^p(\mathbb{T})$, $1 < p < \infty$ is greater than or equal to the $L^\infty(\mathbb{T})$ norm of its symbol. In 1988, A. Böttcher, N. Krupnik, and B. Silbermann posed a question on whether or not the equality holds in the case of continuous symbols. We answer this question in the negative. On the other hand, we show that the essential norm of a Toeplitz operator with a continuous symbol is less than or equal to twice the $L^\infty(\mathbb{T})$ norm of the symbol and prove more precise $p$-dependent estimates.
Hardy Spaces and Toeplitz operators on the Pentablock
School of Mathematics and Statistics, Wuhan University of Technology（武汉理工大学数学与统计学院）；Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore（印度统计研究所班加罗尔统计与数学单元）
For the open unit disc $\mathbb{D}$ in the complex plane, it is well known that if $ϕ\in C(\overline{\mathbb{D}})$ then its Berezin transform $\widetildeϕ$ also belongs to $C(\overline{\mathbb{D}})$. We say that $\mathbb{D}$ is BC-regular. In this paper we study BC-regularity of some pseudoconvex domains in $\mathbb{C}^n$ and show that the boundary geometry plays an important role. We also establish a relationship between the essential norm of an operator in a natural Toeplitz subalgebra and its Berezin transform.
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 …
This paper is a continuation of our study of a class of Toeplitz-like operators with a rational symbol which has a pole on the unit circle. A description of the spectrum and its various parts, i.e., point, residual and continuous spectrum, is given, as well as a description of the essential spectrum. In this case, the essential spectrum need not be connected in ${\mathbb C}$. Various examples illustrate the results.
We prove that every multiplier M (bounded operator commuting with the shift operator) on a large class of Banach spaces of sequences on Z is associated to a function essentially bounded by the norm of M on the spectrum of S.
We show that an infinite Toeplitz+Hankel matrix $T(φ) + H(ψ)$ generates a bounded (compact) operator on $\ell^p(\mathbb{N}_0)$ with $1\leq p\leq \infty$ if and only if both $T(φ)$ and $H(ψ)$ are bounded (compact). We also give analogous characterizations for Toeplitz+Hankel operators acting on the reflexive Hardy spaces. In both cases, we provide an intrinsic characterization of bounded operators of Toeplitz+Hankel form similar to the Brown-Halmos theorem. In addition, we establish estimates for the norm and the essential norm of such operators.
Let $L^2$ be the Lebesgue space of square-integrable functions on the unit circle. We show that the injectivity problem for Toeplitz operators is linked to the existence of geodesics in the Grassmann manifold of $L^2$. We also investigate this connection in the context of restricted Grassmann manifolds associated to $p$-Schatten ideals and essentially commuting projections.
The geometric descriptions of the (essential) spectra of Toeplitz operators with piecewise continuous symbols are among the most beautiful results about Toeplitz operators on Hardy spaces $H^p$ with $1<p<\infty$. In the Hardy space $H^1$, the essential spectra of Toeplitz operators are known for continuous symbols and symbols in the Douglas algebra $C+H^\infty$. It is natural to ask whether the theory for piecewise continuous symbols can also be extended to $H^1$. We answer this question in negative and show in particular that the Toeplitz operator is never bounded on $H^1$ if its symbol has a…
Quantization and spectral properties of Toeplitz operators acting on spaces of pluriharmonic functions over bounded symmetric domains and $\mathbb C^n$ are discussed. Results are presented on the asymptotics \begin{align*} \| T_f^λ\|_λ&\to \| f\|_\infty\\ \| T_f^λT_g^λ- T_{fg}^λ\|_λ&\to 0\\ \| \fracλ{i} [T_f^λ, T_g^λ] - T_{\{f,g\}}^λ\|_λ&\to 0 \end{align*} for $λ\to \infty$, where the symbols $f$ and $g$ are from suitable function spaces. Further, results on the essential spectrum of such Toeplitz operators with certain symbols are derived.
In this paper we consider the C*-algebra $C^{*}(\{C_φ\}\cup\mathcal{T}(PQC(\mathbb{T})))/K(H^{2})$ generated by Toeplitz operators with piece-wise quasi-continuous symbols and a composition operator induced by a parabolic linear fractional non-automorphism symbol modulo compact operators on the Hilbert-Hardy space $H^{2}$. This C*-algebra is commutative. We characterize its maximal ideal space. We apply our results to the question of determining the essential spectra of linear combinations of a class of composition operators and Toeplitz operators.
Let $φ$ be a linear-fractional, non-automorphism self-map of $\mathbb{D}$ that fixes $ζ\in \mathbb{T}$ and satisfies $φ^{\prime}(ζ) \neq 1$ and consider the composition operator $C_φ$ acting on the Hardy space $H^2(\mathbb{D}).$ We determine which linear-fractionally-induced composition operators are contained in the unital C$^*$-algebra generated by $C_φ$ and the ideal $\mathcal{K}$ of compact operators. We apply these results to show that $C^*(C_φ, \mathcal{K})$ and $C^*(\mathcal{F}_ζ)$, the unital C$^*$-algebra generated by all composition operators induced by linear-fractional, non-automor…
