What is the large class of operators for which one can define fractional powers? For example, we can consider an operator $A: D(A) \subset X \rightarrow X$, generator of an analytic semigroup on a Banach space $X$. Can we define the powers $(-A)^\alpha$ for $\alpha>0$ without additional assumptions? I found some references with some restrictions on the spectrum of $A$ or on the associated semigroup. I'm wondering if there is a general definition without further assumption. Any reference would be helpful.