arXiv subjects
Publications and source records attributed to G. Stylogiannis.
In this article we consider the generalized integral operators acting on the Hilbert space $H^2$. We characterize when these operators are uniform, strong and weakly asymptotic Toeplitz and Hankel operators. Moreover we completely describe the symbols $g$ for which these operators are essentially Hankel and essentially Toeplitz.
For $f$ analytic on the unit disc let $r_t(f)(z)=f(e^{it}z)$ and $f_r(z)=f(rz)$, rotations and dilations respectively. We show that for $f$ in the Bergman space $A^p$ and $0<α\leq 1$ the following are equivalent. \begin{itemize} \item[(i)] $\n{r_t(f)-f}_{A^p}=\og(|t|^α), \quad t\to 0$, \item[(ii)] $\n{(f')_r}_{A^p} =\og\left (1-r)^{α-1}\right ), \quad r\to 1^{-}$, \item[(iii)] $\n{f_r-f}_{A^p}=\og((1-r)^α),\quad r\to 1^{-}$. \end{itemize} The Hardy space analogues of these conditions are known to be equivalent by results of Hardy and Littlewood and of E. Storozhenko, and in that setting they describe the mean Lipschitz spaces $Λ(p, α)$. On the way, we provide an elementary proof of the equivalence of $(ii)$ and $(iii)$ in Hardy spaces, and show that similar assertions are valid for certain weighted mean Lipschitz spaces.