Matrix representation of Magnetic pseudo-differential operators via tight Gabor frames

In this paper we use some ideas from \cite{FG-97, G-06} and consider the description of Hörmander type pseudo-differential operators on ${\mathbb{R}}^d$ ($d\ge 1$), including the case of the magnetic pseudo-differential operators introduced in \cite{IMP-1, IMP-19}, with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calder{ó}n-Vaillancourt theorem and Beals’ commutator criterion, and also establish local trace-class criteria.