# Seeley's 'Complex Powers of an Elliptic operator'

#### 10.11.2022, 16:00  –  Raum 1.10 Forschungsseminar Differentialgeometrie

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.

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