On $k$-quasi-$(m,n,\mathbf{C})$-isosymmetric operators

Abstract

We study the set of $k$-quasi-$(m,n,\mathbf{C})$-isosymmetric operators. This family extends the set of $(m,,n,\mathbf{C})$-isosymmetric operators. In the present article, we give operator matrix representation of $k$-quasi-$(m,,n,\mathbf{C})$-isosymmetric operator in order to obtain some structural properties for such operators. We show that if ${\bf R}$ is a $k$-quasi-$(m,n,\mathbf{C})$ isosymmetric, then ${\bf R}^{q}$ is a $k$-quasi-$(m,n,\mathbf{C})$-isosymmetric operator. We show that the product of a $k_1$-quasi-$(m_1,n_1,\mathbf{C})$-isosymmetric and a $k_2$-quasi-$(m_2,n_2,\mathbf{C})$-isosymmetric which are $\mathbf{C}$-double commuting is a $\max\{k_1 ,k_2\}$-quasi-$(m_1+m_2 - 1,n_1+n_2-1, \mathbf{C})$-isosymmetry under suitable conditions. In particular, we prove the stability of perturbation of $k$-quasi-$(m,n, \mathbf{C})$-isosymmetric operator by a nilpotent operator of order $p$ under suitable conditions. Moreover, we give some results about the joint approximate spectrum of a $k$-quasi-$(m,n,\mathbf{C})$-isosymmetric operator.

Keywords

• $m$-isometries
• $n$-quasi-$m$-isometries
• $m$-isometric tuple
• joint approximate spectrum

Supporting Institution

the Deanship of Scientific Research at Jouf University

Project Number

(DSR-2021-03-0337)

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