Annales Academić Scientiarum Fennicć
Mathematica
Volumen 37, 2012, 319-338

OPERATOR-WEIGHTED COMPOSITION OPERATORS BETWEEN WEIGHTED SPACES OF VECTOR-VALUED ANALYTIC FUNCTIONS

José Bonet, M. Carmen Gómez-Collado, David Jornet, and Elke Wolf

Universidad Politécnica de Valencia, Instituto Universitario de Matemática Pura y Aplicada
Edificio 8E, Cubo F, 4^a Planta, E-46022 Valencia, Spain; jbonet 'at' mat.upv.es

Universidad Politécnica de Valencia, Instituto Universitario de Matemática Pura y Aplicada
Edificio 8E, Cubo F, 4^a Planta, E-46022 Valencia, Spain; mcgomez 'at' mat.upv.es

Universidad Politécnica de Valencia, Instituto Universitario de Matemática Pura y Aplicada
Edificio 8E, Cubo F, 4^a Planta, E-46022 Valencia, Spain; djornet 'at' mat.upv.es

University of Paderborn, Mathematical Institute
Warburger Str. 100, D-33098 Paderborn, Germany; lichte 'at' math.uni-paderborn.de

Abstract. We investigate several properties of operator-weighted composition maps W_psi,phi : f \mapsto psi(f \circ phi) on unweighted H(D,X) and weighted H_v^\infty(D,X) spaces of vector-valued analytic functions on the unit disc D. Here phi is an analytic self-map of D and psi is an analytic operator-valued function on D. We characterize when the operator is continuous, maps a neighbourhood into a bounded set or maps bounded sets into relatively compact sets. In this way we extend results due to Laitila and Tylli for the case of Banach valued functions. This more general setting permits us to compare the results in the unweighted and weighted cases. New examples are provided, especially when the spaces X and Y are Köthe echelon spaces. They show the differences between the present setting and the case of functions taking values in Banach spaces.

2010 Mathematics Subject Classification: Primary 47B33; Secondary 46E40, 46A04, 46A45, 46A08, 47B07, 47B38.

Key words: Composition operators, weighted spaces of analytic functions, spaces of vector-valued analytic functions, locally convex spaces, Köthe echelon spaces.

Reference to this article: J. Bonet, M.C. Gómez-Collado, D. Jornet and E. Wolf: Operator-weighted composition operators between weighted spaces of vector-valued analytic functions. Ann. Acad. Sci. Fenn. Math. 37 (2012), 319-338.

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doi:10.5186/aasfm.2012.3723