On Generalized Fredholm Operators in a Right Quaternionic Hilbert Space

The purpose of this paper, is to study and investigate a larger class of operators than the Fredholm ones, is the so-called, generalized Fredholm operators, in a right quaternionic separable Hilbert space with a left multiplication defined on it. More explicitly, we first prove some auxiliary results , in the quaternionic setting, that are needed in the development of this paper. After that, we prove that some special classes of operators are generalized Fredholm ones, and we establish a characterization of the generalized Fredholm operators in a specific quotient algebra. The obtained results leads to the study of the invariance of the generalized Fredholm S-spectrum of a bounded right linear operator defined on a right quaternionic separable Hilbert space under a finite-rank commuting operator.