A Structural Study of Generalized [m,C]-Symmetric Extension Operators

This manuscript introduces and investigates a new class of operators, termedn-quasi-[m,C]-symmetric operators, which generalize and extend the existing notions of [m,C]-symmetric and n-quasi-[m,C]-isometric operators. Specifically, given a conjugation C on a Hilbert space, an operator Q∈B(K) is said to be n-quasi-[m,C]-symmetric if it satisfies the relationQ∗n∑0≤j≤m(−1)jmjCQm−jCQjQn=0. Our study systematically explores the algebraic properties and structural characterization of n-quasi-[m,C]-symmetric operators through matrix representations, providing a deeper understanding of their internal structure. Moreover, we establish sufficient conditions under which the powers and products of such operators inherit the n-quasi-[m,C]-symmetric property. Additionally, we investigate the tensor products of n-quasi-[m,C]-symmetric operators. Finally, we identify conditions that distinguish n-quasi-[m,C]-symmetric operators from n-quasi-[m−1;C]-symmetric operators.