On the spectral properties of the differential operators with involution

In this paper we deal with the problems of the eigenfunction expansions related to the differential operators with involution. The mean value formula for the eigenfunction is obtained with application of the transformation methods of the operators in the symmetric regions. The obtained formula is applied to estimate the eigenfunctions of the given differential operator in the ball. For domains with smooth boundary, the solution to these differential operator problems involves eigenfunction expansions associated with biharmonic-type operator with Navier boundary conditions.