@article{Li2023, 
author = {Chen-Yu Li},
title = {Fractional resolvent family generated by normal operators},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {10},
pages = {23815-23832},
keywords = {fractional resolvent family, normal operator, spectral mapping theorem, stable resolvent},
url = {https://www.sciopen.com/article/10.3934/math.20231213},
doi = {10.3934/math.20231213},
abstract = {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          α        (  t  )  } if    α  ≠  1; and we show that if    {      S          α        (  t  )  } is stable, then we can determine the decay rate by    σ  (  A  ) and some examples are given; we also prove that        S          α        (  t  )  x has a continuous derivative of order    α  β  >  0 if and only if    x  ∈  D  (  I  −  A      )          β      . The main method we used here is the resolution of identity corresponding to a normal operator    A and spectral measure integral.}
}