Holomorphic calculus, numerical range and time discretization of PDE

The holomorphic calculus plays a prominent part as for the theory 
of time dependent problems than for the analysis of their time 
discretizations.
For instance the semigroup theory is based on the rational approximation
$$ e^{-z}=\lim_{n\to \infty}(1+z/n)^{-n},$$ which is nothing else that 
the approximation of the differential equation $u'+u=0$ by the backward 
Euler method.

The talk will be mainly concerned with some new developments in hilbertian 
operator theory, which have interesting applications in the stability study 
of time discretization schemes.