Traces and determinants of linear operators

A general theory of traces tr_DA and determinants det_D(I + A) in normed algebras D of operators acting in Banach spaces B is proposed. In this approach trace and determinant are defined as continuous extensions of the corresponding functionals from finite dimensional operators. We characterize the algebras for which such extensions exist and describe sets of possible values of traces and determinants for the same operator in different algebras. In spite of the fact that the extended traces and determinants may differ in different algebras D , operator I + A (with A ∈ D) is invertible in B if and only if det_D(I + A) does not vanish. Cramer's rule and formulas for the resolvent are obtained and they are expressed in different algebras by the same formulas via det_D(I + A) and tr_D(A). A large set of examples and illustrations are also presented.