NORMAL, COHYPONORMAL AND NORMALOID WEIGHTED COMPOSITION OPERATORS ON THE HARDY AND WEIGHTED BERGMAN SPACES

Abstract

If <TEX>${\psi}$</TEX> is analytic on the open unit disk <TEX>$\mathbb{D}$</TEX> and <TEX>${\varphi}$</TEX> is an analytic self-map of <TEX>$\mathbb{D}$</TEX>, the weighted composition operator <TEX>$C_{{\psi},{\varphi}}$</TEX> is defined by <TEX>$C_{{\psi},{\varphi}}f(z)={\psi}(z)f({\varphi}(z))$</TEX>, when f is analytic on <TEX>$\mathbb{D}$</TEX>. In this paper, we study normal, cohyponormal, hyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces. First, for some weighted Hardy spaces <TEX>$H^2({\beta})$</TEX>, we prove that if <TEX>$C_{{\psi},{\varphi}}$</TEX> is cohyponormal on <TEX>$H^2({\beta})$</TEX>, then <TEX>${\psi}$</TEX> never vanishes on <TEX>$\mathbb{D}$</TEX> and <TEX>${\varphi}$</TEX> is univalent, when <TEX>${\psi}{\not\equiv}0$</TEX> and <TEX>${\varphi}$</TEX> is not a constant function. Moreover, for <TEX>${\psi}=K_a$</TEX>, where |a| < 1, we investigate normal, cohyponormal and hyponormal weighted composition operators <TEX>$C_{{\psi},{\varphi}}$</TEX>. After that, for <TEX>${\varphi}$</TEX> which is a hyperbolic or parabolic automorphism, we characterize all normal weighted composition operators <TEX>$C_{{\psi},{\varphi}}$</TEX>, when <TEX>${\psi}{\not\equiv}0$</TEX> and <TEX>${\psi}$</TEX> is analytic on <TEX>$\bar{\mathbb{D}}$</TEX>. Finally, we find all normal weighted composition operators which are bounded below.