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Orthogonally additive holomorphic functions of bounded type over C(K)

Part of: Linear function spaces and their duals Measures, integration, derivative, holomorphy

Published online by Cambridge University Press:  12 August 2010

Daniel Carando ,
Silvia Lassalle  and
Ignacio Zalduendo
Daniel Carando
Affiliation:
Departamento de Matemática, Pabellón I, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina (dcarando@dm.uba.ar; slassall@dm.uba.ar)
Silvia Lassalle
Affiliation:
Departamento de Matemática, Pabellón I, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina (dcarando@dm.uba.ar; slassall@dm.uba.ar)
Ignacio Zalduendo
Affiliation:
Departamento de Matemática, Universidad Torcuato Di Tella, Miñones 2177 (C1428ATG), Buenos Aires, Argentina (nacho@utdt.edu)
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Abstract

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It is known that all k-homogeneous orthogonally additive polynomials P over C(K) are of the form

Thus, x ↦ xk factors all orthogonally additive polynomials through some linear form μ. We show that no such linearization is possible without homogeneity. However, we also show that every orthogonally additive holomorphic function of bounded type f over C(K) is of the form

for some μ and holomorphic h : C (K) → L1(μ) of bounded type.

Keywords

orthogonally additiveholomorphic functions over C(K)integral representation

MSC classification

Secondary: 46G20: Infinite-dimensional holomorphy 46E15: Banach spaces of continuous, differentiable or analytic functions 46E50: Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 53 , Issue 3 , October 2010 , pp. 609 - 618
Copyright
Copyright © Edinburgh Mathematical Society 2010

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