Subject 
Spectral theory 
Title 
About an approach to the fourthorder operator bundles 
Author(s) 
Abdizhahan Sarsenbi, Makhmud Sadybekov 
Keywords 
differential operator with involution, differential equation with involution, eigenfunctions, biorthogonal decomposition basis 
Abstract 
We consider the bundle form [1]
in the space , where α is a complex number different from zero and ± 1, λ is spectral parameter. The expression (1) should be considered together with the spectral problem [2]
Here, α ≠ 0, α ≠ ± 1 is a complex constant.
The first term in equation (2) contains involution. We assume that the spectral problem (2) is considered with some boundary conditions, the form of which we have not listed yet. There is following relation between the eigenfunctions of the spectral problem (2) and the operator bundle (1).
Theorem 1. The eigenfunctions of the spectral problem (2) are the eigenfunctions of the operator bundle (1).
We consider the fourthorder operator bundle (1) with boundary conditions
Theorem 2. The system eigenfunctions of the bundle (1), (3) contains subsystem
which forms Riesz basis in the space .
