The main focus of this paper is on the relationship between the spectrum of generators and the regularity of the fractional resolvent family. We will give a counter-example to show that the point-spectral mapping theorem is not valid for $\{S_{\alpha }(t)\}$ if $\alpha \ne 1$; and we show that if $\{S_{\alpha }(t)\}$ is stable, then we can determine the decay rate by $\sigma (A)$ and some examples are given; we also prove that $S_{\alpha }(t)x$ has a continuous derivative of order $\alpha \beta >0$ if and only if $x\in D(I-A{)}^{\beta }$. The main method we used here is the resolution of identity corresponding to a normal operator $A$ and spectral measure integral.