Algebras of holomorphic mappings in (DF)-spaces

Let U be an absolutely convex open subset of a complex barrelled (DF)-space E and let F be a commutative Banach algebra with identity. Let Hb(U, F) be the space of holomorphic mappings from U into F that are bounded on the U-bounded sets and let Hb (U, F) be the space of the holomorphic mappings from U into F that are uniformly weakly continuous on the U-bounded sets, both endowed with the topology τb of uniform convergence on the U-bounded sets. The spectra of (Hwu (U, F), τb) and (Hb(U, F), τb) are studied.