Right inverse of the convolution operator in the space of entire functions of exponential type

The present note considers the convolution operators in the space of entire functions of an exponential type less than $\sigma$, $\sigma\leq\infty$. It is shown that a continuous linear right inverse to the convolution operator exists if and only if the characteristic function of the operator has a finite number of zeros in the open circle with the center at zero and radius $\sigma$. Previously, the existence of a continuous linear right inverse for the convolution operator has been studied for spaces of holomorphic functions in a convex domain, germs of holomorphic functions on convex compact sets of entire functions of the order less than $\rho$, $\rho>1$ only. This issue has not been considered for the space of entire functions of an exponential type.