Spectral properties of (m;n)-isosymmetric multivariable operators

Abstract

Inspired by recent works on $m$-isometric and $n$-symmetric multivariables operators on Hilbert spaces, in this paper we introduce the class of $(m,n)$-isosymmetric multivariables operators. This new class of operators emerges as a generalization of the $m$-isometric and $n$-isosymmetric multioperators. We study this class of operators and give some of their basic properties. In particular, we show that if $\mathbf{R}\in {\mathcal{B}}^{(d)}(\mathcal{H})$ is an $(m,n)$-isosymmetric multioperators and $\mathbf{Q}\in {\mathcal{B}}^{(d)}(\mathcal{H})$ is an $q$-nilpotent multioperators, then $\mathbf{R}+\mathbf{Q}$ is an $(m+2q-2,n+2q-1)$-isosymmetric multioperators under suitable conditions. Moreover, we give some results about the joint approximate spectrum of an $(m,n)$-isosymmetric multioperators.