Convolution equations for ultradifferentiable functions and ultradistributions

We characterize surjectivity of convolution operators on spaces of ultradifferentiable functions and ultradistributions of Beurling type in the spirit of Hörmander's convexity conditions. This completes results of Bonet, Galbis, and Meise. In contrast to the classical approach our proofs only use properties of ultradistributions and functional analytic tools.     

Convolution equations and spaces of ultradifferentiable functions

We study spaces of “approximate solutions” to a given convolution equation. In particular we show how their quasianalyticity can be reduced to the study of a suitably constructed Cauchy problem for related spaces. We give a unicity theorem for this problem. This paper is dedicated to the author’s son, Alessandro. The author has been partially supported by Ministero P.I., and G.N.S.A.G.A. of Consiglio Nazionale delle Ricerche.     

Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

In this paper we give global characterisations of Gevrey–Roumieu and Gevrey–Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace–Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their α-duals in the sense of Köthe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.     

Superposition operators between weighted spaces of analytic functions

Abstract

We study the existence and the continuity of superposition operators between weighted spaces of holomorphic functions in terms of the weights.     

On weighted inductive limits of spaces of ultradifferentiable functions and their duals

In the first part of this paper we discuss the completeness of two general classes of weighted inductive limits of spaces of ultradifferentiable functions. In the second part we study their duals and characterize these spaces in terms of the growth of convolution averages of their elements. This characterization gives a canonical way to define a locally convex topology on these spaces and we give necessary and sufficient conditions for them to be ultrabornological. In particular, our results apply to spaces of convolutors for Gelfand–Shilov spaces.     

Algebras of polynomials on locally convex spaces of ultradifferentiable functions and generalized derivation

We consider the space      

Weighted composition operators between weighted spaces of vector-valued holomorphic functions on Banach spaces

Let B(E, F) be the Banach Space of all continuous linear operators from a Banach Space E into a Banach space F. Let U_X and U_Y be balanced open subsets of Banach spaces X and Y, respectively. Let V and W be two Nachbin families of continuous weights on U_X and U_Y, respectively. Let HV(U_X, E) (or HV₀(U_X, E)) and HW(U_Y, F) (or HW₀(U_Y, F)) be the weighted spaces of vector-valued holomorphic functions. In this paper, we investigate the holomorphic mappings ? : U_Y → U_X and ψ : U_Y → B(E, F) which generate weighted composition operators between these weighted spaces.     

Properties of representations of operators acting between spaces of vector-valued functions

A well-known result going back to the 1930s states that all bounded linear operators mapping scalar-valued L ¹-spaces into L ^∞-spaces are kernel operators and that in fact this relation induces an isometric isomorphism between the space of such operators and the space of all bounded kernels. We extend this result to the case of spaces of vector-valued functions. A recent result due to Arendt and Thomaschewski states that the local operators acting on L ^p -spaces of functions with values in separable Banach spaces are precisely the multiplication operators. We extend this result to non-separable dual spaces. Moreover, we relate positivity and other order properties of the operators to corresponding properties of the representations.     

On Borel’s theorem for spaces of ultradifferentiable functions of mean type

Spaces of ultradifferentiable functions of mean type defined by a weight function ω are considered. It is obtained a necessary and sufficient condition for ω under which the corresponding space admits an analog of Borel’s theorem.     

Negative results on the Nash-Moser theorem for Köthe sequence spaces and for spaces of ultradifferentiable functions

Counterexamples to the Nash-Moser inverse function theorem are given for Köthe sequence spaces extending a result of Lojasiewicz and Zehnder. As an application also negative results are obtained for spaces of ultradifferentiable functions of Beurling type, in particular for the Gevrey classes.     

Holomorphic approximation of ultradifferentiable functions

Mathematische Annalen -     

Banach-Saks operators on spaces of continuous functions

Summary Let K be a compact Hausdorff space and let E be a Banach space. We denote by C(K, E) the Banach space of all E-valued continuous functions defined on K, endowed with the supremum norm. We study in this paper Banach-Saks operators defined on C(K, E) spaces. We characterize these operators for a large class of compacts K (the scattered ones), or for a large class of Banach spaces E (the superreflexive ones). We also show by some examples that our theorems can not be extended directly.Partially supported by C.A.I.C.Y.T. grant 0338-84. The author wishes to thank Professor F.Bombal for his encouragement.