ON PROJECTIONAL ORTHOGONAL BASIS OF A LINEAR NON-SELF -ADJOINT OPERATOR

Аннотация

In this paper we study spectral properties of a linear non-self-adjoint operator with an internal
symmetry of the form:
;∗ܮܳ ൌܳ ܮ ,ܲ∗ܮ ൌ ܮܲ
where ܲ∗ ൌ ܲ, ܳ∗ ൌ ܳ are orthogonal projections, ܮ ∗is an operator, adjoint to the operator ܮ in the Hilbert space ܪ .
It is shown that a spectrum of such operator is real. In the case of a discrete operator, with a complete system of
eigenvectors and associated vectors, the projections of eigenvalues and associated vectors of the operator L and its
adjoint operator form an orthonormal basis. A class of Sturm-Liouville operators with such symmetry is found,
moreover, it is found that the characteristic function of such an operator factorizes. An illustrative example is
provided.