Right inverse for a convolution operator in space of germs of analytic functions on connected subsets of $\mathbb C$

Several general results concerning the existence of a continuous linear right inverse (CLRI) of a continuous linear operator are established. Using these results it is possible to obtain first (in a more general situation) necessary and then sufficient conditions (and in several cases, a test) for the existence of a CLRI in spaces of analytic germs on certain classes of connected sets for the convolution operator $L_b$ whose symbol $b(z)$ is an entire function of order 1 and minimal type.