Periodicity in Banach algebras

In this paper, we consider operators that are compositions of an isometric isomorphism and a left multiplier on a Banach algebra, and we provide necessary and sufficient conditions for these operators to have a dense set of periodic elements. As an application of this result, we characterize generalized weighted shifts with a dense set of periodic elements on the standard Hilbert module over C*-algebra of compact operators on a separable Hilbert space. As another application, we characterize generalized weighted shifts with a dense set of periodic elements on the standard Hilbert module over commutative non-unital C*-algebra.