# Completness of roots elementes of linear operators in Banach spaces and application

## TL;DR

This paper investigates the spectral properties of linear operators in Banach spaces, establishing conditions for the completeness of root elements of Schatten class operators and applying these results to boundary value problems with non-self-adjoint operators.

## Contribution

It generalizes known Hilbert space results to Banach spaces and provides new criteria for spectral completeness in this broader context.

## Key findings

- Sufficient conditions for root element completeness in Banach spaces
- Discreteness of spectrum for non-self-adjoint differential operators
- Application to boundary value problems with variable coefficients

## Abstract

In this paper the general spectral properties of linear operators in Banach spaces are studied. We find sufficient conditions on structure of Banach spaces and resolvent properties that guarantee completeness of roots elements of Schatten class operators. This approach generalizes the well known result for operators in Hilbert spaces. In application, the boundary value problems for the abstract equation of second order with variable coefficients are studied. The principal part of the appropriate differential operator is not self-adjoint. The discreetness of spectrum and completeness of root elements of this operator are obtained.