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Trieu Le

Publications and source records attributed to Trieu Le.

At least 19 recordsLinked to original sources
arXiv2024

On compactness of products of Toeplitz operators

We study compactness of product of Toeplitz operators with symbols continuous on the closure of the polydisc in terms of behavior of the symbols on the boundary. For certain classes of symbols $f$ and $g$, we show that $T_fT_g$ is compact if and only if $fg$ vanishes on the boundary. We provide examples to show that for more general symbols, the vanishing of $fg$ on the whole polydisc might not imply the compactness of $T_fT_g$. On the other hand, the reverse direction is closely related to the zero product problem for Toeplitz operators on the unit disc, which is still open.

math.FA↗
arXiv2023

Closed-range posinormal operators and their products

We focus on two problems relating to the question of when the product of two posinormal operators is posinormal, giving (1) necessary conditions and sufficient conditions for posinormal operators to have closed range, and (2) sufficient conditions for the product of commuting closed-range posinormal operators to be posinormal with closed range. We also discuss the relationship between posinormal operators and EP operators (as well as hypo-EP operators), concluding with a new proof of the Hartwig-Katz Theorem, which characterizes when the product of posinormal operators on $\CC^n$ is posinormal.

math.FA↗
arXiv2022

Analogues of Finite Blaschke Products as Inner Functions

We give a generalization of the notion of finite Blaschke products from the perspective of generalized inner functions in various reproducing kernel Hilbert spaces. Further, we study precisely how these functions relate to the so-called Shapiro--Shields functions and shift-invariant subspaces generated by polynomials. Applying our results, we show that the only entire inner functions on weighted Hardy spaces over the unit disk are multiples of monomials, extending recent work of Cobos and Seco.

math.FA↗
arXiv2022

A generalization of the Brown-Halmos theorems for the unit ball

In this paper we generalize the classical theorems of Brown and Halmos about algebraic properties of Toeplitz operators to Bergman spaces over the unit ball in several complex variables. A key result, which is of independent interest, is the characterization of summable functions $u$ on the unit ball whose Berezin transform can be written as a finite sum $\sum_{j}f_j\,\bar{g}_j$ with all $f_j, g_j$ being holomorphic. In particular, we show that such a function must be pluriharmonic if it is sufficiently smooth and bounded. We also settle an open question about $\mathcal{M}$-harmonic functions. Our proofs employ techniques and results from function and operator theory as well as partial differential equations.

math.FA↗
arXiv2020

Finite Rank Perturbations of Toeplitz Products on the Bergman Space

In this paper we investigate when a finite sum of products of two Toeplitz operators with quasihomogeneous symbols is a finite rank perturbation of another Toeplitz operator on the Bergman space. We discover a noncommutative convolution $\diamond$ on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if $F_j, G_j$ ($1\leq j\leq N$) are polynomials of $z$ and $\bar{z}$ then $\sum_{j=1}^{N}T_{F_j}T_{G_j}-T_{H}$ is a finite rank operator for some $L^{1}$-function $H$ if and only if $\sum_{j=1}^{N}F_j\diamond G_j$ belongs to $L^1$ and $H=\sum_{j=1}^{N}F_j\diamond G_j$. In the case $F_j$'s are holomorphic and $G_j$'s are conjugate holomorphic, it is shown that $H$ is a solution to a system of first order partial differential equations with a constraint.

math.FA↗
arXiv2020

Hyponormal Toeplitz Operators on Weighted Bergman Spaces

We consider operators acting on a Hilbert space that can be written as the sum of a shift and a diagonal operator and determine when the operator is hyponormal. The condition is presented in terms of the norm of an explicit block Jacobi matrix. We apply this result to the Toeplitz operator with specific algebraic symbols acting on certain weighted Bergman spaces of the unit disk and determine when such operators are hyponormal.

math.CA↗
arXiv2019

Inner functions in weighted Hardy spaces

Inner functions play a central role in function theory and operator theory on the Hardy space over the unit disk. Motivated by recent works of C. Bénéteau et al. and of D. Seco, we discuss inner functions on more general weighted Hardy spaces and investigate a method to construct analogues of finite Blaschke products.

math.FA↗
arXiv2019

Decomposing algebraic m-isometric tuples

We show that any m-isometric tuples of commuting operators on a finite dimensional Hilbert space can be decomposed as a sum of a spherical isometry and a commuting nilpotent tuple. Our approach applies as well to tuples of algebraic operators that are hereditary roots of polynomials in several variables.

math.FA↗
arXiv2016

On the structure of $\mathcal{N}_p$-spaces in the ball

We study the structure of $\mathcal{N}_p$-spaces in the ball. In particular, we show that any such space is Moebius-invariant and for $0<p \leq n$, all $\mathcal{N}_p$-spaces are different. Our results will be of important uses in the study of operator theory on $\mathcal{N}_p$-spaces.

math.FA↗
arXiv2015

Adjoints of linear fractional composition operators on weighted Hardy spaces

It is well known that on the Hardy space $H^2(\mathbb{D})$ or weighted Bergman space $A^2_α(\mathbb{D})$ over the unit disk, the adjoint of a linear fractional composition operator equals the product of a composition operator and two Toeplitz operators. On $S^2(\mathbb{D})$, the space of analytic functions on the disk whose first derivatives belong to $H^2(\mathbb{D})$, Heller showed that a similar formula holds modulo the ideal of compact operators. In this paper we investigate what the situation is like on other weighted Hardy spaces.

math.FA↗
arXiv2013

Toeplitzness of composition operators in several variables

Motivated by the work of Nazarov and Shapiro on the unit disk, we study asymptotic Toeplitzness of composition operators on the Hardy space of the unit sphere in C^n. We extend some of their results but we also show that new phenomena appear in higher dimensions.

math.FA↗
arXiv2012

Self-adjoint, unitary, and normal weighted composition operators in several variables

We study weighted composition operators on Hilbert spaces of analytic functions on the unit ball with kernels of the form $(1- )^{-γ}$ for $γ>0$. We find necessary and sufficient conditions for the adjoint of a weighted composition operator to be a weighted composition operator or the inverse of a weighted composition operator. We then obtain characterizations of self-adjoint and unitary weighted composition operators. Normality of these operators is also investigated.

math.FA↗
arXiv2011

Boundedness and compactness of composition operators on Segal-Bargmann spaces

For $E$ a Hilbert space, let $\mathcal{H}(E)$ denote the Segal-Bargmann space (also known as the Fock space) over $E$, which is a reproducing kernel Hilbert space with kernel $K(x,y)=\exp(< x,y>)$ for $x,y$ in $E$. If $ϕ$ is a mapping on $E$, the composition operator $C_ϕ$ is defined by $C_ϕh = h\circϕ$ for $h\in \mathcal{H}(E)$ for which $h\circϕ$ also belongs to $\mathcal{H}(E)$. We determine necessary and sufficient conditions for the boundedness and compactness of $C_ϕ$. Our results generalize results obtained earlier by Carswell, MacCluer and Schuster for finite dimensional spaces $E$.

math.FA↗
arXiv2010

Toeplitz operators on Bergman spaces of polyanalytic functions

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.

math.FA↗
arXiv2010

Compact Hankel operators on generalized Bergman spaces of the polydisc

We show that for $f$ a continuous function on the closed polydisc $\bar{\mathbb{D}^n}$ with $n\geq 2$, the Hankel operator $H_{f}$ is compact on the Bergman space of $\mathbb{D}^n$ if and only if there is a decomposition $f=h+g$, where $h$ is in the ball algebra and $g$ vanishes on the boundary of the polydisc.

math.FA↗
arXiv2010

Toeplitz operators on generalized harmonic Bergman spaces

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.

math.FA↗
arXiv2009

Commutant Lifting for Commuting Row Contractions

If $T= \big[ T_1 ... T_n\big]$ is a row contraction with commuting entries, and the Arveson dilation is $\tilde T= \big[ \tilde T_1 ... \tilde T_n\big]$, then any operator $X$ commuting with each $T_i$ dilates to an operator $Z$ of the same norm which commutes with each $\tilde T_i$.

math.OA↗