In this paper we introduce a new class of operators which will be called the class of k-quasi-class A(s, t) operators. An operator T ∈ B(H) is said to be k-quasi-class A(s, t) if T*k((|T*|t|T|2s|T*|t)1t+s−|T*|2t)Tk≥0,where s > 0, t > 0 and k is a natural number. We show that an algebraically k-quasi-class A(s, t) operator T is polaroid, has Bishop’s property β and we prove that Weyl type theorems for k-quasi-class A(s, t) operators. In particular, we prove that if T^* is algebraically k-quasi-class A(s, t), then the generalized a-Weyl’s theorem holds for T. Using these results we show that T^* satisfies generalized the Weyl’s theorem if and only if T satisfies the generalized Weyl’s theorem if and only if T satisfies Weyl’s theorem. We also examine the hyperinvariant subspace problem for k-quasi-class A(s, t) operators.