Finite rank little Hankel operators on $L_{a}^{2}(mathbb{U}_{+})$

Abstract

Let $psiin L^{infty}(mathbb{U_{+}}),$ where $mathbb{U_{+}}$ is the upper half plane in $mathbb{C}$ and $S_{psi}$ be the little Hankel operator with symbol $psi$ defined on the Bergman space $L_{a}^{2}(mathbb{U}_{+}).$ In this paper we have shown that if $S_{psi}$ is of finite rank then $psi=varphi+chi,$ where $chiin left(overline{L_{a}^{2}(mathbb{U}_{+})}right)^{perp}bigcap L^{infty}(mathbb{U}_{+})$ and $overline{varphi}$ is a linear combination of $d_{overline{w}}, win mathbb{U}_{+}$ and some of their derivatives.