Abstract: An operator T is said to be k-quasi-∗-class A if , where k is a natural number. Let denote either the generalized derivation or the elementary operator , where and are the left and right multiplication operators defined on by and respectively. This article concerns some spectral properties of k-quasi-∗-class A operators in a Hilbert space, as the property of being hereditarily polaroid. We also establish Weyl-type theorems for T and , where T is a k-quasi-∗-class A operator and A, are also k-quasi-∗-class A operators. MSC:47B47, 47A30, 47B20, 47B10.