Biorthogonal Expansions in Root Functions of Differential Operators

We consider the problem on the convergence of biorthogonal expansions in a system of eigenfunctions and associated functions for a wide class of operators, whose special cases include nonself-adjoint differential operators. We introduce the notion of almost basis property of systems of root functions of a linear operator. We demonstrate the necessity to use a new method, earlier introduced by the authors, for defining associated functions.