Operators between different weighted Fréchet and (LB)-spaces of analytic functions

Abstract

We study some classical operators defined on the weighted Bergman Frechet space $A^p_{\alpha+}$ (resp. weighted Bergman (LB)-space $A^p_{\alpha-}$) arising as the projective limit (resp. inductive limit) of the standard weighted Bergman spaces into the growth Frechet space $H^\infty_{\alpha+}$ (resp. growth (LB)-space $H^\infty_{\alpha-}$), which is the projective limit (resp. inductive limit) of the growth Banach spaces. We show that, for an analytic self map $\varphi$ of the unit disc $\mathbb{D}$, the continuities of the weighted composition operator $W_{g,\varphi}$, the Volterra integral operator $T_g$, and the pointwise multiplication operator $M_g$ defined via the identical symbol function are characterized by the same condition determined by the symbol's state of belonging to a Bloch-type space. These results have consequences related to the invertibility of $W_{g,\varphi}$ acting on a weighted Bergman Frechet or (LB)-space. Some results concerning eigenvalues of such composition operators $C_\varphi$ are presented.