On a Property of Bases in Banach and Hilbert Spaces

In this work we consider the problem of conservation of the bases’ in Banach spaces after small perturbations for the purpose of applying obtained results to investigation of spectral expansions associated with differential operators. Having known asymptotics of the system of Eigen and adjoin functions for differential or pseudodifferential operators, it is possible to indicate its basisness in Banach space.