In this paper, we review the approach presented by An and Heo on the study of Weyl-type theorems for self-adjoint operators on Krein spaces and show that this approach is not appropriate due to a fallacy. Motivated by this fact, we define a new modification of the kernel of a bounded linear operator on a Krein space, namely $J$-kernel, which allows us to successfully introduce a Fredholm theory in this context and study some variations of Weyl-type theorems for bounded linear operators defined on these spaces. In addition, we will describe the $J$-index in terms of solution sets of homogeneous equations.
OJITO, DANILO POLO; SANABRIA, JOSE; and BUELVAS, YINA OSPINO
"A Fredholm theory on Krein spaces and its application to Weyl-type theorems and homogeneous equations,"
Turkish Journal of Mathematics: Vol. 46:
7, Article 8.
Available at: https://journals.tubitak.gov.tr/math/vol46/iss7/8