Ascent, descent and roots of Fredholm operators
Let $T$ be a Fredholm operator on a Banach space. Say $T$ is rootless if there is no bounded linear operator $S$ and no positive integer $m\geq 2$ such that $S^m=T$. Criteria and examples of rootlessness are given. This leads to a study of ascent and descent whether finite or infinite for $T$ with examples having infinite ascent and descent.