Determinants and Plemelj-Smithies formulas

We establish Plemelj-Smithies formulas for determinants in different algebras of operators. In particular we define a Poincar\'e type determinant for operators on the torus $\Tn$ and deduce formulas for determinants of periodic pseudo-differential operators in terms of the symbol. On the other hand, by applying a recently introduced notion of invariant operators relative to fixed decompositions of Hilbert spaces we also obtain formulae for determinants with respect to the trace class. The analysis makes use of the corresponding notion of full matrix-symbol. We also derive explicit formulas for determinants associated to elliptic operators on compact manifolds, compact Lie groups, and on homogeneous vector bundles over compact homogeneous manifolds.