ag.algebraic geometry – How do I take the n-th root of a linear operator?

Suppose I have a vector space $V/overline{mathbb Q}$ over some algebraically closed field and a semi-simple operator $T$ on it with eigenvalues $alpha_1,dots,alpha_r$ such that $|alpha_i| = 1$ under every complex embedding.

Is there a nice (functorial?) way to define a vector space $V^{1/n}$ of dimension $n$ with an operator $T^{1/n}$ with eigenvalues $alpha_1^{1/n},dots,alpha_r^{1/n}$ (where we take all possible roots)?

(Motivation: I came across Abelian varieties over finite fields where the corresponding Frobenius operators have the above relation.)