On the classes of $n$-Nadir's and $n^*$-Nadir's operators on Hilbert spaces

Abstract

The main aim of this paper is to present some new results for Nadir's operator $N=AB^{*}-BA^{*}$, where $A$ and $B$ are two bounded linear operators. We introduce a generalization of this concept, that is, the $n$-Nadir's and $n^*$-Nadir's operators, where $n$ is a positive integer. We present fundamental properties of these operators, including compactness and normality. Additionally, we establish relationships between these classes, we show that the adjoint of $n$-Nadir's operator is $n^*$-Nadir's operator. Furthermore, if $T$ is an $n$-Nadir's operator, such that $ A $ and $ B $ are two bounded commuting normal operators, then $T$ is an $n^*$-Nadir's operator. Moreover, we study these classes in the special case $n = 2$, with some interesting examples. Other related results are also established.