Around a von Neumann theorem

Michel Crouzeix

IRMAR, Université de Rennes I, Campus de Beaulieu, F-35042 Rennes Cedex, France

michel.crouzeix@univ-rennes1.fr

http://www.maths.univ-rennes1.fr/~crouzeix/

Contributed talk

The following theorem provides a powerfull tool for the numerical analysis of time dependent problems. First we precise the framework

• $H$ is a complex Hilbert space
• $S$ is a closed convex sector or a closed strip of the complex plane
• $A$ is a unbounded closed operator on $H$, with domain $D(A)$ densely included in $H$, moreover there exists some $\zeta \in \mathbb C$ such that $\zeta - A$ is an isomorphism from $D(A)$ onto $H$.

Theorem. We assume that the function $f$ is continuous and bounded in $S$ and holomorphic in the interior of $S$. We assume also that

$$\forall \,v \in D(A), \qquad \langle Av,v\rangle\in S.$$

Then $f(A)$ is a bounded linear operator on $H$ and we have the estimate

$$\Vert f(A)\Vert _{_{H\to H}} \leq (2+{2\over \sqrt 3})\, \sup_{z\in D} \vert f(z)\vert.$$

Notice that this result would follow in a straightforward way from the spectral theory if the operator $A$ would be normal (in this case we could even replace $2+2/\sqrt 3$ by 1). The interest of this theorem is that it coud be used for non normal operators as those which occur in convection-diffusion problem. We prove this result and apply it to stability and convergence analysis of time discretizations of parabolic equations.