Exploring the spectral properties of multivariable $(m,n)$-isosymmetric operators

Drawing from recent advancements in the study of m-isometric and n-symmetric completely positive operators on Hilbert spaces, this paper introduces the concept of $(m; n)$-isosymmetric multivariable operators. This new class of operators serves as a generalization of both m-isometric and n-isosymmetric multioperators. We explore the fundamental properties of these operators, demonstrating that if ${\bf \large R} \in \mathcal{B}^{(d)}(\mathcal{H})$ is an $(m. n)$-isosymmetric multioperator and $\mathcal{Q} \in \mathcal{B}^{(d)}(\mathcal{H})$ is a $q$-nilpotent multioperators, then the sum ${\bf \large R} + \mathcal{Q}$ is an $(m + 2q - 2; n + 2q - 2)$-isosymmetric multioperator under appropriate conditions. Additionally, we present results concerning the joint approximate spectrum of $(m,n)$-isosymmetric multioperators.