Holomorphy types and spaces of entire functions of bounded type on banach spaces

Hypercyclicity of convolution operators on spaces of entire functions

“…More precisely, we use the π 1 -π 2 -holomorphy types introduced by the third and fourth authors in [11]. Although we already knew that π 1 -π 2 -holomorphy types could be used in this context, it was only reading [6] that we realized that the original definitions could be refined (see Definition 2.5) to prove such general results on the hypercyclicity of convolution operators on spaces of entire functions.…”

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

“…To prove this we need that the holomorphy type has some properties, more specifically we need that be a π 1 -holomorphy type. This concept was introduced in [3] and we remember it below.…”

“…Still in this line, Mujica [15] proved that the Borel transform establishes an algebraic isomorphism between the dual of the space H σ (p)b (E) of σ (p)-nuclear entire functions of bounded type defined on E and the space Exp τ (p) (E ) of absolutely τ (p)-summing entire functions of exponential type defined on E . The concept of π 1 -holomorphy type is introduced in [3] in order to generalize all these results. There it is proved that if is a π 1 -holomorphy type, then the Borel transform establishes an algebraic isomorphism between the dual of the space H b (E) of -holomorphy type entire functions of bounded type defined on E and the space Exp (E ) of entire functions of -exponential type defined on E .…”

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

“…The techniques of [4] are a refinement of a general method introduced in [22] to prove existence and approximation results for convolution equations defined on the space H Θb (E) of all entire functions of Θ-bounded type defined on a complex Banach space E.…”

“…The investigation of existence and approximation results for convolution equations was initiated by Malgrange [36] and developed by several authors (see, for instance [11,12,13,18,19,20,21,22,23,31,32,37,38,39,40,46,47,52]).…”