Direct and inverse abstract Cauchy problems with fractional powers of almost sectorial operators

We show the explicit solution operator of an abstract Cauchy problem involving a time-variable coefficient and a fractional power of an almost sectorial operator. Also, by using the representation of the solution operator, we solve the inverse abstract Cauchy problem for recovering the time-variable coefficient. As a complement, we also study similar problems by using almost sectorial operators that depend on a time-variable.