Structure of k -Quasi- m , n -Isosymmetric Operators

Abstract

The investigation of new operators belonging to some specific classes has been quite fashionable since the beginning of the century, and sometimes it is indeed relevant. In this study, we introduce and study a new class of operators called $k$ -quasi- $\left(m,n\right)$ -isosymmetric operators on Hilbert spaces. This new class of operators emerges as a generalization of the $\left(m,n\right)$ -isosymmetric operators. We give a characterization for any operator to be $k$ -quasi- $\left(m,n\right)$ -isosymmetric operator. Using this characterization, we prove that any power of an $k$ -quasi- $\left(m,n\right)$ -isosymmetric operator is also an $k$ -quasi- $\left(m,n\right)$ -isosymmetric operator. Furthermore, we study the perturbation of an $k$ -quasi- $\left(m,n\right)$ -isosymmetric operator with a nilpotent operator. The product and tensor products of two $k$ -quasi- $\left(m,n\right)$ -isosymmetries are investigated.