Concerning Strong Generation of L(ε)

Abstract

Denote by e the locally convex space of entire functions in one complex variable, endowed with the compact-open topology, and by L(e) — the algebra of all continuous linear maps from e into itself. Our main result states that L(e) is strongly generated by the semigroup T(s), s ≥ 0, of translations \((T_{(s)}x)(\zeta)\ =\ x(\zeta\ +\ s),\ x\ \epsilon\ \varepsilon,\ \zeta\ \epsilon\ {\cal C}\) and by the operator Tz, given by\(T_zx=zx, where\ z(\zeta)=\zeta\).