arXiv subjects
Publications and source records attributed to Aissa Bouhali.
We investigate the commutant problem for Toeplitz operators on the Bergman space of the unit disk whose symbols belong to a subclass of biharmonic functions. We obtain a complete characterization of when two such Toeplitz operators commute. As a consequence, we derive a full description of normal Toeplitz operators with symbols in this class.
A major open problem in the Theory of Toeplitz operators on the analytic Bergman space over the unit disk is the characterization of the commutant of a given Toeplitz operator--that is, the set of all bounded Toeplitz operators that commute with it. In this paper, we provide a complete description of bounded Toeplitz operators $T_f$, where the symbol $f$ has a truncated polar decomposition, that commute with a Toeplitz operator whose symbol is the sum of a quasihomogeneous function and a bounded analytic function.
A major open question in the theory of Toeplitz operator on the Bergman space of the unit disk of the complex plane is the complete characterization of the set of all Toeplitz operators that commute with a given operator. In \cite{al}, the authors showed that when a sum $S=T_{e^{imθ}f}+T_{e^{ilθ}g}$, where $f$ and $g$ are radial functions, commutes with a sum $T=T_{e^{ipθ}r^{(2M+1)p}}+T_{e^{isθ}r^{(2N+1)s}}$, then $S$ must be of the form $S=cT$, where $c$ is a constant. In this article, we will replace $r^{(2M+1)p}$ and $r^{(2N+1)s}$ with $r^n$ and $r^d$, where $n$ and $d$ are in $\mathbb{N}$, and we will show that the same result holds.
A major open question in the theory of Toeplitz operator on the Bergman space of the unit disk of the complex plane is to fully characterize the set of all Toeplitz operators that commute with a given one. In [2], the second author described the sum $S = T_{e^{imθ}f} +T_{e^{ilθ}g}$, where $f$ and $g$ are radial functions, that commutes with the sum $T = T_{e^{ipθ}r(2M+1)p} +T_{e^{isθ}r(2N+1)s}$ . It is proved that $S = cT$, where $c$ is a constant. In this article, we shall replace $r^{(2M+1)p}$ and $r^{(2N+1)s}$ by $r^n$ and $r^d$ respectively, with $n$ and $d$ in $\mathbb{N}$, and we shall show that the same result holds.