The author establishes some weaker characteristic conditions of the operator \(t_c\) and investigates invariant operators. It is shown that if an operator \(T\) can commute with one of the operators \(\sigma_p\) (\(p\neq 0\) and a root of unity), then \(T\) is an invariant operator. It is pointed out that the set of all invariant operators is a ring which is an algebra isomorphism with the ring of sequences of constants. A reverse result of the expansion theorem is also presented. The paper ends with presenting an open problem.