The results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y, respectively. Our first main result is an error equality for all equations of the class A∗Ax + x = f or in mixed formulation A∗y + x = f, Ax = y, where the exact solution (x, y) is in D(A) × D(A∗). Here A is a linear, densely defined and closed (usually a differential) operator and A∗its adjoint. In this paper we deal with very conforming mixed approximations, i.e., we assume that the approximation (x, y) belongs to D(A) × D(A∗). In order to obtain the exact global error value of this approximation one only needs the problem data and the mixed approximation itself, i.e., we have the equality |x - x|² + |A(x - x)|² + |y - y|² + |A∗ (y - y)|² = M(x, y), where M(x, y) := |f - x - A∗y|² + |y - Ax|² contains only known data. Our second main result is an error estimate for all equations of the class A∗Ax + ix = f or in mixed formulation A∗y + ix = f, Ax = y, where i is the imaginary unit. For this problem we have the two-sided estimate (equation presented) where M_i(x, y) := |f - ix - A∗y|² + |y - Ax|² contains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.