Fractional resolvent family generated by normal operators

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.