Fréchet algebras generated by certain of their elements

We consider F-algebras A that are generated by elements of the form z, (z−λ1e)−1,…,(z−λNe)−1, where e is the identity. If A has no topclogical divisors of zero we show that A is isomorphic to H(Ω), where Ω is finitely connected region. We also study F-algebras in which {e,z,z−1,z2,z−2,…} is basis.