Slice Monogenic Functions and Their Functional Calculi
Speaker:Prof. Fabrizio Colombo
Associate Professor
Politecnico di Milano
Date & Time:3 Oct 2013 (Thursday) 10:30
Venue:J317
Organized by:Department of Mathematics

Abstract

In this talk we will give an overview of the theory of slice monogenic functions and of their functional calculi. The first functional calculus we presents is the so called S-functional calculus which works for n-tuples of bounded, but also for unbounded, not necessarily commuting operators. The S-functional calculus is based on the notion S-spectrum, which naturally arises from the definition of the S-resolvent operator for n-tuples of operators. The S-resolvent operator plays the same role that the usual resolvent operator plays for the Riesz-Dunford functional calculus. In the case one considers commuting operators (bounded or unbounded) there is the possibility to simplify the computation of the S-spectrum. In fact, we can use the so-called F-spectrum, which, in this case, equals the S-spectrum. The main good property of the F-spectrum, is that it is easier to compute than the S-spectrum. In case of commuting operators the S-functional calculus, based on the F-spectrum, will be called SC-functional calculus. The F-spectrum is related to the F-functional calculus which is based on the integral version of the Fueter mapping theorem.

Biography

Prof. Fabrizio Colombo is currently an Associate Professor of Mathematical Analysis in Politecnico di Milano in Italy. Prof. Colombo obtained his Ph.D. degree in Mathematics in Universita degli Studi di Milano. He is interested in Inverse problems for parabolic integrodifferential equations, Spectral theory for n-tuples of operators and for quaternionic operators, Elliptic and Parabolic equations, Semigroup Theory, Hypercomplex Analysis and its Applications, and Inverse and control problems. The recent publications of Prof. Colombo include, On some operators associated to superoscillations, On the Cauchy problem for the Schrodinger equation with superoscillatory initial Data, On two approaches to the Bergman theory for slice regular functions, The inverse Fueter mapping theorem in integral form using spherical monogenics, Some remarks on the S-spectrum etc.