Resumen de Weighted composition operators on spaces of analytic funcions

Esther Gomez Orts

• The aim of this thesis is to study some properties of the weighted composition operators on different weighted spaces of analytic functions.

  Given a weight v strictly positive and continuous on the complex disc, we consider certain Banach spaces of analytic functions on the complex disc. These spaces are the sets of the holomorphic functions on the disc f such that the supremum, when z is in the disc, of v(z)|f(z)| is finite. We also consider the spaces of the holomorphic functions f such that v(z)|f(z)| tends to 0 whenever |z| goes to 1.

  Given a sequence of weights, we work with the spaces described by the intersection or union of the weighted Banach spaces determined by the weights in the sequence. The space of the intersection is a Fréchet space and it is the projective limit of the mentioned Banach spaces. This space is endowed with the projective limit topology. The space given by the union is an LB-space (limit of Banach), and it is the inductive limit of the mentioned spaces, with the inductive limit topology. When the sequence is given by the weights (1-|z|)^n with n natural, the space of the union is called Korenblum space, which is also an inductive limit.

  In the thesis we study the continuity, compactness and invertibility of the weighted composition operators on the spaces described above. We also study some properties of the spectrum and point spectrum.