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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=(vn) 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 Cg: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.