Weakly compact composition operators between vector valued weighted spaces of holomorphic functions
by
José Bonet
Universidad Politécnica de Valencia, Spain
Coauthors: Miguel Friz (Universidad Politécnica de Valencia, Spain)

Let E be a complete, barrelled locally convex space, let V=(v_n) be an increasing sequence of radial, continuous weights on the unit disc D of the complex plane, and let g be an analytic self map on D. We characterize when the composition operator C_g:f --> f o g from the vector valued weighted space HV(D, E) of holomorphic functions on D into itself maps bounded sets into relatively weakly compact subsets (i.e. it is reflexive), and when it maps a zero-neighborhood into a relatively weakly compact set (i.e., it is weakly compact). The case of one weight and a Banach space E was treated by Bonet, Doma\'nski and Lindström in [3].

The proof requires the following ingredients: (1) a study of reflexive and weakly compact wedge operators T --> LTR between spaces of continuous linear operators between complete, barrelled spaces (see Saksman and Tylli [4] for the case of Banach spaces), (2) a representation of the space HV(D, E) as a space of operators which complements the results of Bierstedt and Holtmanns [1], and (3) the extension of the characterizations of [2] for systems of weights.

References:

1. K.D. Bierstedt, S. Holtmanns, An operator representation for weighted inductive limits of spaces of vector valued holomorphic functions, Results Math. 36 (1999) 9-20.

2. J. Bonet, P. Doma\'nski, M. Lindström, Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions, Canad. Math. Bull. 42 (1999) 139-148.

3. J. Bonet, P. Doma\'nski, M. Lindström, Weakly compact composition operators on weighted vector valued Banach spaces of analytic mappings, Ann. Acad. Sci. Fenn. Math. (to appear).

4. E. Saksman, H. Tylli, Weak essential spectra of multiplication operators on spaces of bounded linear operators, Math. Ann. 299 (1994) 299-309.

Date received: February 14, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafv-24.