  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.
