In this paper, we prove that every norm closed, commutative, semisimple, strictly cyclic algebra A ⊂ B(X) is reflexive. This extends a result of A. Lambert to the case of Banach spaces. We also prove that such algebras are hereditarily reflexive. In [2] A. Lambert introduced the notion of strictly cyclic operator algebras. He gave examples of strictly cyclic algebras, generated by certain weighted shifts on Hilbert spaces (Donaghue algebras) as well as non-singly generated abelian strictly cyclic algebras. Subsequently, other authors [1, 4] investigated the invariant subspace lattices of such algebras. Some of these strictly cyclic algebras are semi-simple, commutative algebras [2]. Lambert [2] proved that such algebras, A, are reflexive (i.e. for every operator T ∈ B(H), such that TK ⊂ K for every closed subspace K ⊂ H that is invariant for A, it follows that T ∈ A.) In this paper, we extend this result to strictly cyclic algebras of operators on Banach spaces. We prove, in addition, that every semi-simple, strictly-cyclic, commutative algebra on a Banach space is hereditarily reflexive, which is a new result for Hilbert spaces as well. Our proofs are more simple than Lambert’s and use only elementary properties of semisimple Banach algebras.