A class of weighted composition operators whose Range and Null spaces are complemented

Abstract

In this paper, we prove that the null space of a weighted composition operator on $\ell_p~ (1 \leq p < \infty)$ is a complemented subspace. We also give a necessary and sufficient condition for a weighted composition operator on $\ell_p$ whose range space is of finite co-dimension. Thereafter, we characterize a class of weighted composition operators whose range space is a complemented subspace. Lastly, we characterize weighted composition operators on $\ell_p$ which are Fredholm operators.