# Lecture 13

### From 15th ISeminar 2011/12 Operator Semigroups for Numerical Analysis

### **Rational Approximations and Analytic Semigroups**

Dear Participants,

as you might remember from last week, rational approximations are special time discretisation methods, where stability and consistency play an important role. It was possible to extend the scalar estimates to the case of multiplication operators in case we have a geometric condition on those operators: its spectrum lies in a sector. Hence, it is natural to ask whether the same holds for general sectorial operators.

Is is our aim this week to show that this is indeed the case. Unfortunately, the matter is more technical now and we have to develop a kind-of functional calculus which will enable us to do so. You can read more on this in the lecture following the PDF link below.

We kindly ask everyone to send in their solutions.

Have fun and good reading,

your virtual lecturers

Alexander

András

Bálint

Petra

ISEM15_Lecture13.pdf

**Discussion** of Lecture 13.

Solutions and discussion of the **exercises**.

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