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

We study some classical operators defined on the weighted Bergman Frechet space A(alpha+)(p) (resp. weighted Bergman (LB)-space A(alpha-)(p)) arising as the projective limit (resp. inductive limit) of the standard weighted Bergman spaces into the growth Frechet space H-alpha+(infinity) (resp. growth (LB)-space H-alpha-(infinity)), which is the projective limit (resp. inductive limit) of the growth Banach spaces. We show that, for an analytic self map phi of the unit disc D, the continuities of the weighted composition operator W-g,W-phi, 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,W-phi acting on a weighted Bergman Frechet or (LB)-space. Some results concerning eigenvalues of such composition operators C-phi are presented.