On a new class of operators related to quasi-Fredholm operators

Abstract

In this paper, we introduce a generalization of quasi-Fredholm operators [7] to $k$-quasi-Fredholm operators on Hilbert spaces for nonnegative integer $k$. The case when $k = 0,$ represents the set of quasi-Fredholm operators and the meeting of the classes of $k$-quasi-Fredholm operators is called the class of pseudo-quasi-Fredholm operators. We present some fundamental properties of the operators belonging to these classes and, as applications, we prove some spectral theorem and finite-dimensional perturbations results for these classes. Also, the notion of new index of a pseudo-quasi-Fredholm operator called $pq$-index is introduced and the stability of this index by finite-dimensional perturbations is proved. This paper extends some results proved in [5] to closed unbounded operators.