Olga Aryasova (Inst. of Geophysics, Nat. Acad. of Sciences of Ukraine / Friedrich–Schiller–Univ. Jena)
Alberto Richtsfeld (UP)
In this talk, I will review Seeley's groundbreaking paper 'Complex Powers of an Elliptic Operator'. In this paper, he defines complex powers of an elliptic, classical pseudo-differential operator \(A\) with a ray of minimal growth, and shows that these stay in the class of classical pseudo-differential operators. Furthermore, he identifies the poles of the function that maps a complex number \(s\) to the Schwartz-kernel of \(A^s\) and gives an explicit formula for the residues at these poles as well as a formula for the value of this function at the point \(s=0\). I will then shed a light on the numerous applications such as Zeta-functions of pseudo-differential operators, heat kernel asymptotics and Weyl asymptotics for self-adjoint differential operators.