The author provides a well written introduction to some recent developments of the theory of semigroups of linear operators in Banach spaces, in particular to functional calculus and kernel estimates. The paper contains some examples illustrating the abstract results and some proofs are exhibited, with a special interest to elliptic and parabolic boundary value problems.
It is divided into eight chapters with the titles: Semigroups, Holomorphic semigroups, Asymptotics, Functional calculus, Form methods and functional calculus, Fourier multipliers and maximal regularity, Gauss estimates and ultracontractivity, Elliptic operators.
For the entire collection see [Zbl 1055.35006].