Generalized eigenvectors of linear operators and biorthogonal systems

Abstract

The notions of a set of generalized eigenvalues and a set of generalized eigenvectors of a linear operator in Euclidean space are introduced. In addition, we provide a method to find a biorthogonal system of a subsystem of eigenvectors of some linear operators in a Hilbert space whose systems of canonical eigenvectors are over-complete. Related to our problem, we will show an example of a linear differential operator that is formally adjoint to Bessel-type differential operators. We also investigate basis properties (completeness, minimality, basicity) of the systems of generalized eigenvectors of this differential operator.

Keywords

• Linear operator
• generalized eigenvector
• Bessel function
• complete system
• minimal system
• biorthogonal system

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