Spectral Attributes of Self-Adjoint Fredholm Operators in Hilbert Space: A Rudimentary Insight

Abstract

In defining the finiteness or infiniteness conditions of discrete spectrum of the Schrodinger operators, a fundamental understanding on $n(1, F(\cdot ))$ 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 $F(\cdot )$ defined in the interval of (a, b) and discontinuous points of the function $n(1, F(\cdot ))$. Besides, the obtained relation allowed us to define the finiteness of the numbers $z\in (a,b)$ for which 1 is an eigenvalue of F(z) even if $F(\cdot )$ is not defined at a and b. Results were validated through some examples.