On a mean value theorem in the class of Herglotz functions and its applications.
The multivariate linear model, in which the matrix of the first order parameters is divided into two matrices: to the matrix of the useful parameters and to the matrix of the nuisance parameters, is considered. We examine eliminating transformations which eliminate the nuisance parameters without loss of information on the useful parameters and on the variance components.
Let P,Q be two linear idempotents on a Banach space. We show that the closedness of the range and complementarity of the kernel (range) of linear combinations of P and Q are independent of the choice of coefficients. This generalizes known results and shows that many spectral properties of linear combinations do not depend on their coefficients.