The purpose of the paper is to introduce and study a class of multivariable operators on semi-Hilbertian spaces, i.e. spaces generated by positive semidefinite sesquilinear forms. Let $\mathbf{T}=(T_1,\dots ,T_m)$ and let A be a commuting m-tuple of operators and a positive operator on a complex Hilbert space, respectively. We introduce an A-hyponormal tuple of operators and study some properties of this class of multivariable operators. An m-tuple of operators $\mathbf{T}=(T_1,\dots ,T_m)\in {\mathcal{B}}_A(\mathcal{H}{)}^m$ is said to be A-hyponormal tuple if $\sum _{1\le i,j\le m}{⟨[T_i^{\mathrm{♯}},T_j]x_i|x_j⟩}_A\ge 0,$ for each finite collections $(x_i{)}_{1\le i\le m}\in \mathcal{H}.$

This manuscript introduces and investigates a new class of operators, termedn-quasi-$[m,\mathcal{C}]$-symmetric operators, which generalize and extend the existing notions of $[m,\mathcal{C}]$-symmetric and n-quasi-$[m,\mathcal{C}]$-isometric operators. Specifically, given a conjugation $\mathcal{C}$ on a Hilbert space, an operator $\mathcal{Q}\in \mathcal{B}\left(\mathcal{K}\right)$ is said to be n-quasi-$[m,\mathcal{C}]$-symmetric if it satisfies the relation${\mathcal{Q}}^{\ast n}\left({\displaystyle \sum _{0\le j\le m}}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{m}{j}}\right)\mathcal{C}{\mathcal{Q}}^{m-j}\mathcal{C}{\mathcal{Q}}^j\right){\mathcal{Q}}^n=0$. Our study systematically explores the algebraic properties and structural characterization of n-quasi-$[m,\mathcal{C}]$-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-$[m,\mathcal{C}]$-symmetric property. Additionally, we investigate the tensor products of n-quasi-$[m,\mathcal{C}]$-symmetric operators. Finally, we identify conditions that distinguish n-quasi-$[m,\mathcal{C}]$-symmetric operators from n-quasi-$[m-1;\mathcal{C}]$-symmetric operators.

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‐(m, n)‐isosymmetric operators on Hilbert spaces. This new class of operators emerges as a generalization of the (m, n)‐isosymmetric operators. We give a characterization for any operator to be k‐quasi‐(m, n)‐isosymmetric operator. Using this characterization, we prove that any power of an k‐quasi‐(m, n)‐isosymmetric operator is also an k‐quasi‐(m, n)‐isosymmetric operator. Furthermore, we study the perturbation of an k‐quasi‐(m, n)‐isosymmetric operator with a nilpotent operator. The product and tensor products of two k‐quasi‐(m, n)‐isosymmetries are investigated.

El Moctar Ould Beiba, n-power-Posinormal Operators. Methods Funct. Anal. Topology, 2021, Vol. 27, no. 1, 18-24

The study of m-tuples of operators was the subject of intensive research by many authors. Our aim in this work is to consider a generalization of the concept of posinormality of a single operator done in Rhaly (Posinormal operators. J Math Soc Jpn. 1994;46(4):587–605) to the concept of joint posinormality for m-tuples of operators on a complex Hilbert space. We study some of the basic properties of posinormal m-tuples of operators.