SMS scnews item created by Daniel Hauer at Mon 18 Mar 2019 1344
Type: Seminar
Distribution: World
Expiry: 1 Apr 2019
Calendar1: 1 Apr 2019 1400-1500
CalLoc1: AGR Carslaw 829
CalTitle1: An operator theoretic approach to pseudo-differential calculus
Auth: dhauer@p635m2.pc (assumed)

# An operator theoretic approach to pseudo-differential calculus

### Pierre Portal

Pierre Portal
Australian National University, Canberra, Australia
Mon 1st Apr 2019, 2-3pm, Carslaw Room 829 (AGR)

## Abstract

Can one use pseudo-differential calculus in situations where no Fourier transform is available? On a manifold, one can use the euclidean pseudo-differential calculus locally, but can one find a global analogue? In the simpler case of (radial) Fourier multiplier calculus, the answer is well known: one can think of this calculus as a functional calculus of the Laplacian, and then generalise it, by operator theoretic methods, to large classes of Laplace like operators acting on various Banach spaces. The quintessential example of this approach is given by Alan McIntosh’s construction of a holomorphic calculus for sectorial operators.

Generalising the full pseudo-differential calculus is more challenging, but an operator theoretic perspective is nonetheless available: the Weyl calculus of the (euclidean) position and momentum operators. In this talk, we see how this calculus can be extended to pairs of group generators acting on a Banach space, and satisfying the same algebraic commutator condition as the position and momentum operators. This provides a foundation for the study of stochastic or geometric analogues of the standard quantum harmonic oscillator $\Delta -|x{|}^{2}$.

This is joint work with Jan van Neerven (Delft).

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