<abstract><p>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 \neq 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 &gt; 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.</p></abstract>