# 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.)