In this thesis we consider continuity and positivity properties of pseudo-differential operators in Gelfand-Shilov and Pilipović spaces, and their distribution spaces. We also investigate composition property of pseudo-differential operators with symbols in quasi-Banach modulation spaces.

We prove that positive elements with respect to the twisted convolutions, possesing Gevrey regularity of certain order at origin, belong to the Gelfand-Shilov space of the same order. We apply this result to positive semi-definite pseudo-differential operators, as well as show that the strongest Gevrey irregularity of kernels to positive semi-definite operators appear at the diagonals.

We also prove that any linear operator with kernel in a Pilipović or Gelfand-Shilov space can be factorized by two operators in the same class. We give links on numerical approximations for such compositions and apply these composition rules to deduce estimates of singular values and establish Schatten-von Neumann properties for such operators.  

Furthermore, we derive sufficient and necessary conditions for continuity of the Weyl product with symbols in quasi-Banach modulation spaces.