Concerning Strong Generation of L(ε)

Abstract

Denote by ε the locally convex space of entire functions in one complex variable, endowed with the compact-open topology, and by L(ε) — the algebra of all continuous linear maps from ε into itself. Our main result states that L(ε) 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 T _z, given by$T_zx=zx, where\ z(\zeta)=\zeta$.