Continuation of a linear operator to an involution operator

A bounded linear operator $A\colon X\to X$ in a linear topological space $X$ is called a $p$-involution operator, $p\ge2$, if $A^p=I$, where $I$ is the identity operator. In this paper, we describe linear $p$-involution operators in a linear topological space over the field $\mathbb C$ and prove that linear operators can be continued to involution operators.