The roots of algebraic linear transformations on infinite dimensional complex vector spaces

Let $F$ be a field, $V$ is a vector space on $F$, $\alpha$ is a linear transformation on $V$ and m is a positive integer greater than 1. If a linear transformation $\chi$ on $V$ satisfies ${\chi }^m=\alpha$, then $\chi$ is called an m-th root of $\alpha$. In this paper, using the Prüfer-Baer theorem of bounded modules over the principal ideal domain, we study the linear transformation which is algebraically on infinite dimensional complex vector spaces, and give the local and global conditions under which an m-th root of $\alpha$ can be expressed as a polynomial in $\alpha$. The obtained results deepen the related results in the matrix analysis.