Sarason Toeplitz product problem asks when the operator TuTv is bounded on various Hilbert spaces of analytic functions, where u and v are analytic. The problem is highly nontrivial for Toeplitz operators on the Hardy space and the Bergman space (even in the case of the unit disk). In this paper, we provide a complete solution to the problem for a class of Fock spaces on the complex plane. In particular, this generalizes an earlier result of Cho, Park, and Zhu.
While the symbol map for the collection of bounded Toeplitz operators is well studied, there has been little work on a symbol map for densely defined Toeplitz operators. In this work a family of candidate symbols, the Sarason Sub-Symbols, is introduced as a means of reproducing the symbol of a densely defined Toeplitz operator. This leads to a partial answer to a question posed by Donald Sarason in 2008. In the bounded case the Toeplitzness of an operator can be classified in terms of its Sarason Sub-Symbols. This justifies the investigation into the application of the Sarason Sub-Symbols on d…
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.
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.
The condition numbers $CN(T)==||T|| .||T^{-1}|| $ of Toeplitz and analytic $n\times n$ matrices $T$ are studied. It is shown that the supremum of $CN(T)$ over all such matrices with $||T|| \leq1$ and the given minimum of eigenvalues $r=min_{i=1..n}|λ_{i}|>0$ behaves as the corresponding supremum over all $n\times n$ matrices (i.e., as $\frac{1}{r{}^{n}}$ (Kronecker)), and this equivalence is uniform in $n$ and $r$. The proof is based on a use of the Sarason-Sz.Nagy-Foias commutant lifting theorem.
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.
By working with all collection of all the Sarason Hilbert Hardy spaces for the annulus algebra an improvement to the results of Aryana and Clancey on eigenvalues of self adjoint Toeplitz operators on an annulus is obtained. The ideas are applied to Toeplitz operators on the Neil algebra. These examples may provide a template for a general theory of Toeplitz operators with respect to an algebra.
This paper contains a further analysis of the Toeplitz-like operators $T_ω$ on $H^p$ with rational symbol $ω$ having poles on the unit circle that were previously studied in [5.6]. Here the adjoint operator $T_ω^*$ is described. In the case where $p=2$ and $ω$ has poles only on the unit circle $\mathbb{T}$, a description is given for when $T_ω^*$ is symmetric and when $T_ω^*$ admits a selfadjoint extension. Also in the case where $p=2$, $ω$ has only poles on $\mathbb{T}$ and in addition $ω$ is proper, it is shown that $T_ω^*$ coincides with the unbounded Toeplitz operator defined by Sarason in…
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 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.
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.
In this paper we characterize the Schatten $p$ class membership of Toeplitz operators with positive measure symbols acting on generalized Fock spaces for the full range $0 < p < \infty$.
In this paper, we investigate the boundedness of Toeplitz product $T_{f}T_{g}$ and Hankel product $H_{f}^{*} H_{g}$ on Fock-Sobolev space for two polynomials $f$ and $g$ in $z,\overline{z}\in\mathbb{C}^{n}$. As a result, the boundedness of Toeplitz operator $T_{f}$ and Hankel operator $H_{f}$ with the polynomial symbol $f$ in $z,\overline{z}\in\mathbb{C}^{n}$ is characterized.
An estimate for the norm of selfadjoint Toeplitz operators with a radial, bounded and integrable symbol is obtained. This emphasizes the fact that the norm of such operator is strictly less than the supremum norm of the symbol. Consequences for time-frequency localization operators are also given.
Let $F^{2,m}(\mathbb{C})$ denote the Fock-Sobolev space of complex plane. In this paper, we characterize the semi-commutator of two Toeplitz operators on $F^{2,m}(\mathbb{C})$ is zero. The result is different from the result of Bauer, Choe, and Koo. (J. Funct. Anal., 268 (2015): 3017-3060.)
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.
Commuting Toeplitz operators with harmonic symbols on the Fock space
School of Mathematics and Statistics, Chongqing Technology and Business University（重庆工商大学数学与统计学院）；School of Mathematical Sciences, Chongqing Normal University（重庆师范大学数学科学学院）
In this paper we will give necessary and sufficient conditions for the operator $T_ν^s$ to be in the symmetrically normed ideal $\mathcal{C}_Φ$ for an arbitrary symmetric norming function $Φ$ where $T_ν$ is the Toeplitz operator on the generalized Fock Space (also known as the generalized Bargmann-Fock space) with a positive measure symbol $ν$ and $0<s\leq 1$.
Our purpose is to characterize the so-called horizontal Fock-Carleson type measures for derivatives of order $k$ (we write it $k$-hFC for short) for the Fock space as well as the Toeplitz operators generated by sesquilinear forms given by them. The boundedness conditions for such operators are found. We introduce real coderivatives of $k$-hFC type measures and show that the C*-algebra generated by Toeplitz operators with the corresponding class of symbols is commutative and isometrically isomorphic to certain $C^*$-subalgebra of $L_{\infty}(\mathbb{R}^{n})$. The above results are extended to m…
In sprite by Berger-Coburn theorems and their conjecture in \cite{Coburn1994}, we use the Fourier transform to decompose $ T_{g}$ as an infinite sum of Toeplitz operators with symbols which have compact support in the frequency domain. As a consequence, we obtain a sufficient condition for $ T_{g}$ to be bounded in terms of the Carleson measure conditions defined by the heat transform of the symbol $g$. Moreover the decomposition of a Toeplitz operator leads us to get easily understanding that for a bounded function $g$, if its Berezin transform vanishes at infinity, then the Toeplitz operator…
The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions. In the context of a reproducing kernel Hilbert space we propose a certain framework for a `maximally possible' extension of the notion of Toeplitz operators for a `maximally wide' class of `highly singular' symbols. Using the language of sesquilinear forms we …
The Fock space $\mathcal{F}(\mathbb{C}^n)$ is the space of holomorphic functions on $\mathbb{C}^n$ that are square-integrable with respect to the Gaussian measure on $\mathbb{C}^n$. This space plays an important role in several subfields of analysis and representation theory. In particular, it has for a long time been a model to study Toeplitz operators. Esmeral and Maximenko showed in 2016 that radial Toeplitz operators on $\mathcal{F}(\mathbb{C})$ generate a commutative $C^*$-algebra which is isometrically isomorphic to the $C^*$-algebra $C_{b,u}(\mathbb{N}_0,ρ_1)$. In this article, we exten…
Given a contraction A on a Hilbert space H, an operator T on H is said to be A-invariant if <Tx,x>=<TAx,Ax> for every x in H such that ||Ax||=||x||. In the special case in which both defect indices of A are equal to 1, we show that every A-invariant operator is the compression to H of an unbounded linear transformation that commutes with the minimal unitary dilation of A. This result was proved by Sarason under the additional hypothesis that A is of class C_{00}, leading to an intrinsic characterization of the truncated Toeplitz operators. We also adapt to our more general context other result…
The Fock space consists of all entire functions which are square integrable with respect to Gauss measure. The Toeplitz algebra is the C*-algebra generated by the Toeplitz operator with bounded symbol on the Fock space. In this paper, we study the Toeplitz algebra on the Fock space.
The framework of the paper is that of the full Fock space ${\Cal F}^2({\Cal H}_n)$ and the Banach algebra $F^\infty$ which can be viewed as non-commutative analogues of the Hardy spaces $H^2$ and $H^\infty$ respectively. An inner-outer factorization for any element in ${\Cal F}^2({\Cal H}_n)$ as well as characterization of invertible elements in $F^\infty$ are obtained. We also give a complete characterization of invariant subspaces for the left creation operators $S_1,\cdots, S_n$ of ${\Cal F}^2({\Cal H}_n)$. This enables us to show that every weakly (strongly) closed unital subalgebra of $\{…
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.
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.
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 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.
Let $f$ and $g$ be functions, not identically zero, in the Fock space $F^2$ of $C_n$. We show that the product $T_fT_{\bar g}$ of Toeplitz operators on $F^2$ is bounded if and only if $f(z)=e^{q(z)}$ and $g(z)=ce^{-q(z)}$, where $c$ is a nonzero constant and $q$ is a linear polynomial.
We prove several results concerning the theory of Toeplitz algebras over $p$-Fock spaces using a correspondence theory of translation invariant symbol and operator spaces. The most notable results are: The full Toeplitz algebra is the norm closure of all Toeplitz operators with bounded uniformly continuous symbols. This generalizes a result obtained by J. Xia (J. Funct. Anal. 269:781-814, 2015) in the case $p = 2$, which was proven by different methods. Further, we prove that every Toeplitz algebra which has a translation invariant $C^\ast$ subalgebra of the bounded uniformly continuous functi…
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 operators in the Fock space, under rather general conditions imposed on the symbols. It is proved that if the operator has finite rank and the symbol is a function then the operator and the symbol should be zero. The method of proving is different from the one used previously for finite rank theorems, and it enables one to get rid of the compact support condition for symbols imposed previously.
In this paper, we study Toeplitz algebras generated by certain class of Toeplitz operators on the $p$-Fock space and the $p$-Bergman space with $1<p<\infty$. Let BUC($\mathbb C^n$) and BUC($\mathbb B_n$) denote the collections of bounded uniformly continuous functions on $\mathbb C^n$ and $\mathbb B_n$ (the unit ball in $\mathbb C^n$), respectively. On the $p$-Fock space, we show that the Toeplitz algebra which has a translation invariant closed subalgebra of BUC($\mathbb C^n$) as its set of symbols is linearly generated by Toeplitz operators with the same space of symbols. This answers a ques…
We study mapping properties of Toeplitz operators $T_μ$ associated to nonnegative Borel measure $μ$ on the complex space $\mathbb{C}^n$. We, in particular, describe the bounded and compact operators $T_μ$ acting between Fock spaces in terms of the objects $t$-Berezin transforms, averaging functions, and averaging sequences of their inducing measures $μ$. An asymptotic estimate for the norms of the operators has been also obtained. The results obtained extend a recent work of Z. Hu and X. Lv and fills the remaining gap when both the smallest and largest Banach--Fock spaces are taken into accoun…
In this paper we show that the C*-algebra generated by radial Toeplitz operators with $L_{\infty}$-symbols acting on the Fock space is isometrically isomorphic to the C*-algebra of bounded sequences uniformly continuous with respect to the square-root-metric $ρ(j,k)=|\sqrt{\vphantom{jk}j}-\sqrt{\vphantom{jk}k}\,|$. More precisely, we prove that the sequences of eigenvalues of radial Toeplitz operators form a dense subset of the latter C*-algebra of sequences.
We study the boundedness of Toeplitz operators on Segal-Bargmann spaces in various contexts. Using Gutzmer's formula as the main tool we identify symbols for which the Toeplitz operators correspond to Fourier multipliers on the underlying groups. The spaces considered include Fock spaces, Hermite and twisted Bergman spaces and Segal-Bargmann spaces associated to Riemannian symmetric spaces of compact type.
We study the asymptotic expansion of the product of two Toeplitz operators on the Fock space. In comparison to earlier results we require significantly less derivatives and get the expansion to arbitrary order. This, in particular, improves a result of Borthwick related to Toeplitz quantization. In addition, we derive an intertwining identity between the Berezin star product and the sharp product.
The well know conjecture of {\it Coburn} [{\it L.A. Coburn, {On the Berezin-Toeplitz calculus}, Proc. Amer. Math. Soc. 129 (2001) 3331-3338.}] proved by {\it Lo} [{\it M-L. Lo, {The Bargmann Transform and Windowed Fourier Transform}, Integr. equ. oper. theory, 27 (2007), 397-412.}] and {\it Englis} [{\it M. Engli$\check{s}$, Toeplitz Operators and Localization Operators, Trans. Am. Math Society 361 (2009) 1039-1052.}] states that any {\it Gabor-Daubechies} operator with window $ψ$ and symbol ${\bf a}(x,ω)$ quantized on the phase space by a {\it Berezin-Toeplitz} operator with window $Ψ$ and sy…
