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A Fredholm theory on Krein spaces and its application to Weyl-type theorems and homogeneous equations

Abstract:

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