Higher-Order Extensions of C-Hyponormality via n-Quasi-C-Hyponormal Operators

In this paper, we investigate generalizations and extensions of C-hyponormal operators, focusing on the class of n-quasi-C-hyponormal operators. We provide a detailed structural analysis, including decomposition results that split an operator into a C-hyponormal part and a nilpotent part. Moreover, we study the behavior of operators in this class under direct sums, tensor products, and tensor sums, highlighting the conditions under which these operations preserve the n-quasi-C-hyponormality property. Illustrative examples are provided to demonstrate both the possibilities and limitations of these extensions. The results offer new insights into the interplay between C-symmetry and quasi-hyponormality, broadening the framework of operator theory in Hilbert spaces.