# Semigroups with an Inner Function Calculus

Project Number: 613.001.004
Project Manager: prof. dr. H.J. Zwart
Faculty of Electrical Engineering, Mathematics and Computer Science
Tel.: +31-53-4893464
Email: h.j.zwart@utwente.nl

Project website: Semigroups with an Inner Function Calculus

## Summary

Functions of operators appear naturally in many places. For instance, the solution of the linear differential equation x'(t)=A x(t), x(0)=x_0 is given by x(t)=exp(At)x_0\$. If the operator A is bounded and the function f is an entire analytic function, then f(A) is easily defined by using the power series of f. However, when A becomes unbounded and/or f is not an entire function, then this power series expansion can no longer be used.

In this project we study a new class of operator-function pairs. We choose A to be the infinitesimal generator of a bounded semigroup, i.e., exp(At) exists and is bounded for t > 0, and f to be an inner function analytic on the left half-plane. That is the mapping s to f(s) is an analytic function for s with real part negative, in this half-plane |f(s)| is bounded by one, and on the imaginary axis |f(s)|=1, almost everywhere. The importance of this class comes from applications to numerical analysis and infinite-dimensional systems theory. The well-known Crank-Nicolson scheme and many other Runge-Kutta methods in numerical analysis can be seen as inner functions applied to A."

Project duration: 1-10-2011 / 1-10-2015
Project budget: 205 k-€ funding
Number of person/months: 1.2 fte / year
Involved groups: Hybrid Systems