Analysis Seminar

Speaker: Flavia Colonna, George Mason University

Abstract: Let $\psi$ be an analytic function on the open unit disk D and let $\phi$ be an analytic self-map of D. The weighted composition operator with symbols $\psi$ and $\phi$ is defined on the space of analytic functions on D as$W_{\psi, \phi} f = \psi \cdot (f\circ \phi)$. Let H be a reproducing kernel Hilbert space of analytic functions on the unit disk. In this talk, we determine conditions on H and its kernel K which allow us to characterize the bounded and the compact weighted composition operators from H into weighted-type Banach spaces. We obtain an exact formula for the operator norm and an approximation of the essential norm of the operators mapping into the space $H_{\mu}^\infty$ consisting of the analytic functions of whose modulus is $O(\mu)$, where the weight $\mu$ is a positive continuous function on D. We obtain an exact formula of the essential norm for a large class of weighted Hardy Hilbert spaces. We also discuss the case when the Hilbert space H is replaced by a general Banach space of analytic functions such that all point evaluations are bounded linear functionals. This is joint work with Maria Tjani.