Spectral attributes of self-adjoint fredholm operators in hilbert space: a rudimentary insight

• Main Authors: Muminov, M. I., Ghoshal, S. K.
• Format: Article
• Published: Birkhauser Verlag AG 2019
• Subjects: QA75 Electronic computers. Computer science
• Online Access: http://eprints.utm.my/id/eprint/88486/; http://www.dx.doi.org/10.1007/s11785-018-0865-7

In defining the finiteness or infiniteness conditions of discrete spectrum of the Schrodinger operators, a fundamental understanding on n(1 , F(·)) is crucial, where n(1, F) is the number of eigenvalues of the Fredholm operator F to the right of 1. Driven by this idea, this paper provided the invertibility condition for some class of operators. A sufficient condition for finiteness of the discrete spectrum involving the self-adjoint operator acting on Hilbert space was achieved. A relation was established between the eigenvalue 1 of the self-adjoint Fredholm operator valued function F(·) defined in the interval of (a, b) and discontinuous points of the function n(1 , F(·)). Besides, the obtained relation allowed us to define the finiteness of the numbers z∈ (a, b) for which 1 is an eigenvalue of F(z) even if F(·) is not defined at a and b. Results were validated through some examples.