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This question was considered by Functional Analysis specialists around 1960 or earlier. In the case of Banach Algebras $C(X)$ (for Hausdorff compact $X$) this gets reduced often to studying the auto-homeomorphisms of $X$, and it is extra interesting when the compact space is nice. In this context, in 1961, I have rediscovered that orientation preserving homeomorphisms of $\,X:=[0;1]\,$ admit square root (and quite a bit more but I didn't know at the time that the full group of homeomorphisms was described already in full by someone else, sometimes earlier). This means that there are root squares (and similar) of the respective operators. I also constructed an orientation preserving homeomorphism of $\,\Bbb S^1\,$ which does not admit a square root hence the respective operator didn't either. The later result was presented in a monograph by Marek Kuczma, on functional equations, (which made me feel good, especially that I was a student at the time).