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

Abstract

We study some classical operators defined on the weighted Bergman Fréchet space A p α+ (resp. weighted Bergman (LB)-space A p α− ) arising as the projective limit (resp. inductive limit) of the standard weighted Bergman spaces into the growth Fréchet space H ∞ α+ (resp. growth (LB)-space H ∞ α− ), which is the projective limit (resp. inductive limit) of the growth Banach spaces. We show that, for an analytic self map φ of the unit disc D , the continuities of the weighted composition operator Wg,φ , the Volterra integral operator Tg , and the pointwise multiplication operator Mg 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 Wg,φ acting on a weighted Bergman Fréchet or (LB)-space. Some results concerning eigenvalues of such composition operators Cφ are presented.

Volterra operators between limits of Bergman-type weighted spaces of analytic functions

“…If the bounded subsets of X are relatively compact, i.e., X is a Schwartz space, then bounded operators coincides with compact operators. By [17,Corollary 3.2], this is the case for A p α+ and A p α− . Therefore, Proposition 2.1 will play a crucial role in characterizing compactness of the Volterra operator T g on A p α+ and A p α− .…”

“…is the Cesàro operator. This operator acting on A p α+ and A p α− was investigated by the author in [17].…”

“…The motivation of the present research is due to the investigation on the spectrum of Volterra-type operators on weighted Bergman spaces carried out by Aleman and Constantin in [5], and the study of the author [17] where continuity, compactness and spectrum of Cesàro operator in the same context of unions and intersections of weighted Bergman spaces considered here inspired by the works of Albanese, Bonet and Ricker [1,3]. The Volterra operator is closely related to Cesàro operator.…”