On a Problem that Does not Have Basis Property of Root Vectors, Associated with a Perturbed Regular Operator of Multiple Differentiation

A spectral problem for a multiple differentiation operator with integral perturbation of boundary value conditions which are regular but not strongly regular is considered in the paper. The feature of the problem is the absence of the basis property of the system of root vectors. A characteristic determinant of the spectral problem is constructed. It is shown that absence of the basis property of the system of root functions of the problem is unstable with respect to the integral perturbation of the boundary value condition

Riesz Inequality for the System of Root Functions of Second Order Ordinary Differential Operator

An ordinary differential operator of second order with coefficients is considered. The Riesz property of the system of root functions of the given operator is studied. The criterion of Bessel property in 2 L , of root functions system is established and use it to obtain sufficient conditions for the Riesz property of a system of normalized root functions of this operator in p L .

Uniform Basis Property of the System of Root Vectors of the Dirac Operator

We study one-dimensional Dirac operator L on the segment [0,π] with regular in the sense of Birkhoff boundary conditions U and complex-valued summable potential P=(pij(x)), i,j=1,2. We prove uniform estimates for the Riesz constants of systems of root functions of a strongly regular operator L assuming that boundary-value conditions U and the number ∫(p1(x)-p4(x))dx are fixed and the potential P takes values from the ball B(0,R) of radius R in the space Lϰ for ϰ>1. Moreover, we can choose the system of root functions so that it consists of eigenfunctions of the operator L except for a finite number of root vectors that can be uniformly estimated over the ball ∥P∥ϰ≤R.