Christian Seifert: On fractional powers of sectorial operators

Abstract

In 2007, Caffarelli and Silvestre published a paper how to obtain the fractional Laplacian, i.e. \((-\Delta)^\alpha\) for some \(\alpha\in (0,1)\), by means of an extension problem. Put differently, the solution of an (abstract) ODE gives rise to the fractional Laplacian. Since then there has been quite a few steps to generalise the approach, first to operators on Hilbert spaces and then to some classes of operators to Banach spaces. In this talk we will explain the method and show that for a general sectorial operator in a Banach space the extension problem can be used to obtain fractional powers.