Abstract: An operator T is called k-quasi-*-A(n) operator, if T^*k|T¹⁺ⁿ|^2/(1+n)T^k≥T^*k|T^*|²T^k, k ∈ Z, which is a generalization of quasi-*-A(n) operator. In this paper we prove some properties of k-quasi-*-A(n) operator, such as, if T is a k-quasi-*-A(n) operator and N(T )N(T^*), then its point spectrum and joint point spectrum are identical. Using these results, we also prove that if T is a k-quasi-*-A(n) operator and N(T )N(T^*), then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum. 

Key words: k-quasi-?-A(n) operator, quasisimilarity, single valued extension property; Weyl spectrum, essential approximate point spectrum