On a class of Toeplitz and little Hankel operators on $L_{a}^{2}(\mathbb{U}_{+})$

Abstract

In this paper we establish certain algebraic properties of Toeplitz operators and a class of little Hankel operators defined on the Bergman space of the upper half plane. We show that if $K$ is a compact\break operator on $L_{a}^{2}(\mathbb{U}_{+}), M(s)=\frac{i-s}{i+s}, \tau_{a}(s)=\frac{(c-1)+sd}{(1+c)s-d}$ where $a=c+id\in\mathbb{D}, s\in \mathbb{U}_{+}$ and $Jf(s)=f(-\overline{s})$ then $\displaystyle\lim_{|a|\rightarrow 1^{-}}||K-T_{J(M\circ\tau_{a})}KT^{*}_{M\circ\tau_{a}}||=0$ and for $\varphi, \psi \in h^{\infty}(\mathbb{D})$, if $\hbar_{\alpha_{s}(\psi\circ M)}T_{\varphi\circ M}-T_{\varphi\circ M}\hbar_{\alpha_{s}(\psi\circ M)}$ is compact, then $\displaystyle\lim_{\substack{w=x+iy\\y\rightarrow 0}}||c([\hbar_{\alpha_{s}(\psi\circ M)}d_{\overline{w}}]\otimes[\hbar^{*}_{\varphi\circ M}d_{w}])+c([\hbar_{J(\varphi\circ M)}d_{\overline{w}}]\otimes[\hbar^{*}_{\alpha_{s}(\psi\circ M)}d_{w}])||\break=0,$ where $d_{\overline{w}}(s)=\dfrac{1}{\sqrt{\pi}}\dfrac{w+i}{\overline{w}-i}\dfrac{(-2i)Im~w}{(s+w)^{2}}, w \in \mathbb{U}_{+}, \hbar_{\varphi}$ is the little Hankel operator on $L_{a}^{2}(\mathbb{U}_{+})$ with symbol $\varphi$ and $\alpha_{s}$ is a function defined on $\mathbb{U}_{+}$ with $|\alpha_{s}|=1$, for all $s\in \mathbb{U}_{+}.$ Applications of these results are also obtained.