ON PROJECTIONAL ORTHOGONAL BASIS OF A LINEAR NON-SELF -ADJOINT OPERATOR - Shaldanbayev A.Sh.,; Shaldanbayeva A.А.,; Shaldanbay B.A., (Academic Scientific Journal of Computer Science, 2019)
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.