Holomorphy types and the Fourier-Borel transform between spaces of entire functions of a given type and order defined on Banach spaces

Abstract

Let E be a Banach space and be a π 1 -holomorphy type. The main purpose of this paper is to show that the Fourier-Borel transform is an algebraic isomorphism between the dual of the space Exp k ,A (E) of entire functions on E of order k and -type strictly less than A and the space Exp

Hypercyclicity, existence and approximation results for convolution operators on spaces of entire functions

“…As done just below of [23,Definition 4.3] we may consider the space Exp k Θ ′ ,A (E), for every k ∈ [1, +∞] and A ∈ (0, +∞], but in this paper we are particularly interest in case A = +∞. We recall the definition now.…”

“…In this work we give contributions in two directions. In the first, we explore hypercyclicity results for convolution operators on the space Exp k Θ,0,A (E), introduced in [23], of Θ entire functions of a given type A and order k on a complex Banach space E, where Θ is a given holomorphy type. These results generalize the hypercyclicity results obtained in [4,6,9,28,35].…”

Duality results in Banach and quasi-Banach spaces of homogeneous polynomials and applications

“…The same problem happens in the investigation of existence and approximation results for convolution equations. This line of investigation was initiated by Malgrange [39] and developed by several authors (see, for instance [14,15,16,22,23,24,25,26,27,33,34,40,41,42,43,45,46,50]). In this context, a result of [26] (refined in [5]) gives a general method to prove existence and approximation results for convolution equations defined on certain spaces of entire functions of bounded type.…”

Convolution operators on spaces of entire functions

“…The study of convolution operators on spaces of entire functions was initiated by Malgrange in the case of entire functions of several complex variables. The case of infinitely many complex variables is more delicate, and the technique based on maps of bounded type introduced by Gupta (1969) became standard and has been used by several authors since then (see ).…”