Abstract. This article deals with the relationship between an operator ideal and its natural polynomial extensions. We define the concept of coherent sequence of polynomial ideals and also the notion of compatibility between polynomial and operator ideals. We study the stability of these properties for maximal and minimal hulls, adjoint and composition ideals. We also relate these concepts with conditions on the underlying tensor norms.

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A result of Godefroy and Shapiro states that the convolution operators on the space of entire functions on C n , which are not multiples of identity, are hypercyclic. Analogues of this result have appeared for some spaces of holomorphic functions on a Banach space. In this work, we define the space holomorphic functions associated to a sequence of spaces of polynomials and determine conditions on this sequence that assure hypercyclicity of convolution operators. Some known results come out as particular cases of this setting. We also consider holomorphic functions associated to minimal ideals of polynomials and to polynomials of the Schatten-von Neumann class.

ABSTRACT. We investigate the summability of the coefficients of m-homogeneous polynomials and m-linear mappings defined on ℓ p -spaces. In our research we obtain results on the summability of the coefficients of m-linear mappings defined on ℓ p 1 × · · · × ℓ p m . The first results in this respect go back to Littlewood and Bohnenblust and Hille (for bilinear and mlinear forms on c 0 ) and Hardy and Littlewood and Praciano-Pereira (for bilinear and m-linear forms on arbitrary ℓ p -spaces). Our results recover and in some case complete these old results through a general approach on vector valued m-linear mappings.

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