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

Document Type : Original Article

Authors

Sworup Kumar Das

Namita Das

Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar, Odisha

10.22034/kjm.2021.243561.1964

Abstract

Let $\psi\in L^{\infty}(\mathbb{U_{+}}),$ 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 $\chi\in \left(\overline{L_{a}^{2}(\mathbb{U}_{+})}\right)^{\perp}\bigcap L^{\infty}(\mathbb{U}_{+})$ and $\overline{\varphi}$ is a linear combination of $d_{\overline{w}}, w\in \mathbb{U}_{+}$ and some of their derivatives.

Keywords

Bergman space

Upper half plane

Little Hankel operators

Finite rank operators

Essentially bounded functions

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