Microlocal and Spectral Analysis of Tensor Products of Pseudodifferential Operators

In this thesis we study tensor products of pseudodifferential operators from the microlocal point of view. More generally, we consider the class of so-called bisingular pseudodifferential operators, introduced by L. Rodino in 1975. In Chapter 1 we recall some preliminaries on tensor products and bisingular operators on the product of two closed manifolds. In Chapter 2 we define a wave front set for such operators, called the bi-wave front set, and analyse its properties. The definition is given using the calculus only, and microlocality and microellipticity results are given. In Chapter 3 we study the asymptotic behavior of the counting function of tensor products of pseudodifferential operators. We obtain, in particular, the sharpness of the remainder term in the corresponding Weyl formulae, which we prove by means of the analysis of some explicit examples. In Chapter 4 we consider a class of global bisingular operators based on Shubin calculus. For such class, we show the equivalence of their ellipticity, defined by the invertibility of certain associated homogeneous principal symbols, and their Fredholm mapping property in associated scales of Sobolev spaces.