Let T be a bounded linear operator on a complex Hilbert space H . For a positive integer k , an operator T is said to be a k -quasi-2-isometric operator if T ^∗k ( T ^∗2 T ² − 2 T ^∗ T + I ) T^k = 0, which is a generalization of 2-isometric operator. In this paper, we consider basic structural properties of k -quasi-2-isometric operators. Moreover, we give some examples of k -quasi-2-isometric operators. Finally, we prove that generalized Weyl’s theorem holds for polynomially k -quasi-2-isometric operators. Let H be an infinite dimensional separable Hilbert space. We denote by B ( H ) the algebra of all bounded linear operators on H , write N ( T ) and R ( T ) for the null space and range space of T , and also, write σ ( T ), σ _a ( T ) and iso σ ( T ) for the spectrum, the approximate point spectrum and the isolated point spectrum of T , respectively. An operator T is called Fredholm if R ( T ) is closed, α ( T ) = dim N ( T ) < ∞ and β ( T ) = dim H/R ( T ) < ∞. Moreover if i ( T ) = α ( T ) − β ( T ) = 0, then T is called Weyl. The Weyl spectrum w ( T ) of T is defined by w ( T ) := { λ ∈ ℂ : T − λ is not Weyl}. Following [12] , we say that Weyl’s theorem holds for T if σ ( T )\ w ( T ) = π ₀₀ ( T ), where π ₀₀ ( T ) := { λ ∈ iso σ ( T ) : 0 < dim N ( T − λ ) < ∞}. More generally, Berkani investigated B -Fredholm theory (see [4 , 5 , 6] ). An operator T is called B -Fredholm if there exists n ∈ ℕ such that R ( Tⁿ ) is closed and the induced operator T _[n] : R ( Tⁿ ) ∋ x → Tx ∈ R ( Tⁿ ) is Fredholm, i.e., R ( T _[n] ) = R ( T ⁿ⁺¹ ) is closed, α ( T _[n] ) < ∞ and β ( T _[n] ) = dim R ( Tⁿ )/ R ( T _[n] ) < ∞. Similarly, a B -Fredholm operator T is called B -Weyl if i ( T _[n] ) = 0. The B -Weyl spectrum σ_BW ( T ) is defined by σ_BW ( T ) = { λ ∈ ℂ : T − λ is not B -Weyl}. We say that generalized Weyl’s theorem holds for T if σ ( T ) \ σ _BW ( T ) = E ( T ), where E ( T ) denotes the set of all isolated points of the spectrum which are eigenvalues (no restriction on multiplicity). Note that, if generalized Weyl’s theorem holds for T , then so does Weyl’s theorem [5] . In [1] Agler obtained certain disconjugacy and Sturm-Lioville results for a subclass of the Toeplitz operators. These results were suggested by the study of operators T which satisfies the equation, Such T are natural generalizations of isometric operators ( T ^∗ T = I ) and are called 2-isometric operators. It is known that an isometric operator is a 2-isometric operator. 2-isometric operators have been studied by many authors and they have many interesting properties (see [2 , 3 , 7 , 8 , 9 , 14] ). In order to extend 2-isometric operators we introduce k -quasi-2-isometric operators defined as follows: Definition 1.1. For a positive integer k , an operator T is said to be a k - quasi -2- isometric operator if It is clear that each 2-isometric operator is a k -quasi-2-isometric operator and each k -quasi-2-isometric operator is a ( k + 1)-quasi-2-isometric operator. In this paper we give a necessary and sufficient condition for T to be a k -quasi-2-isometric operator. Moreover, we study characterizations of weighted shift operators which are k -quasi-2-isometric operators. Finally, we prove polynomially k -quasi-2-isometric operators satisfy generalized Weyl’s theorem. We begin with the following theorem which is the essence of this paper; it is a structure theorem for k -quasi-2-isometric operators. Theorem 2.1. If T^k does not have a dense range, then the following statements are equivalent: (1) T is a k-quasi-2-isometric operator; (2) on , where T ₁ is a 2- isometric operator and Furthermore , σ ( T ) = σ ( T ₁ ) ∪ {0}. Proof . (1) ⇒ (2) Consider the matrix representation of T with respect to the decomposition : Let P be the projection onto . Since T is a k -quasi-2-isometric operator, we have Therefore On the other hand, for any x = ( x ₁ , x ₂ ) ∈ H , we have which implies Since σ ( T )∪ M = σ ( T ₁ )∪ σ ( T ₃ ), where M is the union of the holes in σ ( T ) which happen to be subset of σ ( T ₁ ) ∩ σ ( T ₃ ) by Corollary 7 of [11] , and σ ( T ₁ ) ∩ σ ( T ₃ ) has no interior point and T ₃ is nilpotent, we have σ ( T ) = σ ( T ₁ ) ∪ {0}. (2) ⇒ (1) Suppose that on , where and . Since we have where . It follows that T ^∗k ( T ^∗2 T ² − 2 T ^∗ T + I ) T^k = 0 on . Thus T is a k -quasi-2-isometric operator.               ☐ Corollary 2.2. If is a k-quasi- 2- isometric operator and T ₁ is invertible, then T is similar to a direct sum of a 2- isometric operator and a nilpotent operator. Proof . Since T ₁ is invertible, we have σ ( T ₁ ) ∩ σ ( T ₃ ) = φ . Then there exists an operator S such that T ₁ S − S T ₃ = T ₂ [15] . Since , it follows that Corollary 2.3. If T is a k-quasi -2- isometric operator and R ( T^k ) is dense , then T is a 2- isometric operator. Proof. This is a result of Theorem 2.1.               ☐ Corollary 2.4. If T is a k-quasi- 2- isometric operator, then so is Tⁿ for every natural number n. Proof. We decompose T as Then by Theorem 2.1, . Hence T ₁ is a 2-isometric operator, by [14 , Theorem 2.1], is a 2-isometric operator. Since Tⁿ is a k -quasi-2-isometric operator for every natural number n by Theorem 2.1.               ☐ Lemma 2.5. T is a k-quasi- 2 -isometric operator if and only if for every x ∈ H . Theorem 2.6. Let T be a k-quasi -2- isometric operator and M be an invariant subspace for T. Then the restriction T | _M is also a k-quasi -2- isometric operator. Proof. For x ∈ M , we have Thus T | _M is a k -quasi-2-isometric operator.               ☐ Example 2.7. Given a bounded sequence α : α ₀ , α ₁ , α ₂ , . . . (called weights), the unilateral weighted shift W _α associated with α is the operator on l ₂ defined by W _α e _n = α_ne _n+1 for all n ≥ 0, where is the canonical orthogonal basis for l ₂ and | α_n | ≠ 0 for each n ≥ 0. Then the following statement holds: W_α is a k -quasi-2-isometric operator if and only if where Proof . By calculation, , by definition, W_α is a k -quasi-2-isometric operator if and only if               ☐ Remark 2.8. Let W_α be the unilateral weighted shift with weight sequence ( α_n ) _{n ≥0} and | α_n | ≠ 0 for each n ≥ 0. From Example 2.7 we obtain the following characterizations: 1. W_α is a k -quasi-2-isometric operator if and only if for n ≥ k . 2. {| α_n |} is a decreasing sequence of real numbers converging to 1 for n ≥ k . 3. for n ≥ k + 1. 4. Let 2 = | α_k |, 1 = | α _k+1 | = | α _k+2 | = | α _k+3 | = ⋯. Then W_α is a ( k + 1)-quasi-2-isometric operator but not a k -quasi-2-isometric operator. In the sequel, we focus on polynomially k -quasi-2-isometric operators. We say that T is a polynomially k -quasi-2-isometric operator if there exists a nonconstant complex polynomial p such that p ( T ) is a k -quasi-2-isometric operator. It is clear that a k -quasi-2-isometric operator is a polynomially k -quasi-2-isometric operator. The following example provides an operator which is a polynomially k -quasi-2-isometric operator but not a k -quasi-2-isometric operator. Example 2.9. Let Then T is a polynomially k-quasi- 2- isometric operator but not a k-quasi -2- isometric operator. Proof. Since we have Then Therefore T is not a k -quasi-2-isometric operator. On the other hand, consider the complex polynomial h ( z ) = ( z − 1) ² + 1. Then h ( T ) = I , and hence T is a polynomially k -quasi-2-isometric operator.               ☐ Recall that an operator T is said to be isoloid if every isolated point of σ ( T ) is an eigenvalue of T and polaroid if every isolated point of σ ( T ) is a pole of the resolvent of T . In general, if T is polaroid, then it is isoloid. However, the converse is not true. Theorem 2.10. Let T be a polynomially k-quasi -2 -isometric operator. Then T is polaroid. Proof . We first show that a k -quasi-2-isometric operator is polaroid. We consider the following two cases: Case I: If the range of T^k is dense, then T is a 2-isometric operator, T is polaroid. Since an invertible 2-isometric operator is a unitary operator by [2 , Proposition 1.23], and if T is a non-invertible 2-isometric operator, then iso σ ( T ) is empty. Case II: If the range of T^k is not dense, by Theorem 2.1, we have Let λ ∈ iso σ ( T ). Suppose that T ₁ is a non-invertible 2-isometric operator. Then σ ( T ) = D , where D is the closed unit disk. Since σ ( T ) = σ ( T ₁ ) ∪ {0}, we have iso σ ( T ) is empty; thus T ₁ is a invertible 2-isometric operator and λ ∈ iso σ ( T ₁ ) or λ = 0, T ₁ is a unitary operator, T ₃ is nilpotent. It is easy to prove that T − λ has finite ascent and descent, i.e., λ is a pole of the resolvent of T , therefore T is polaroid. Next we show that a polynomially k -quasi-2-isometric operator is polaroid. If T is a polynomially k -quasi-2-isometric operator, then p ( T ) is a k -quasi-2-isometric operator for some nonconstant polynomial p . Hence it follows from the first part of the proof that p ( T ) is polaroid. Now apply [10 , Lemma 3.3] to conclude that p ( T ) polaroid implies T polaroid.               ☐ Corollary 2.11. Let T be a polynomially k-quasi- 2- isometric operator. Then T is isoloid. An operator T is said to has the single valued extension property (abbreviated SVEP) if, for every open subset G of ℂ, any analytic function f : G → H such that ( T − z ) f ( z ) ≡ 0 on G , we have f ( z ) ≡ 0 on G . Theorem 2.12. Let T be a polynomially k-quasi -2 -isometric operator. Then T has SVEP. Proof . We first suppose that T is a k -quasi-2-isometric operator. We consider the following two cases: Case I: If the range of T^k is dense, then T is a 2-isometric operator, T has SVEP by [8 , Theorem 2]. Case II: If the range of T^k is not dense, by Theorem 2.1, we have Suppose ( T − z ) f ( z ) = 0, f ( z ) = f ₁ ( z ) ⊕ f ₂ ( z ) on . Then we can write And T ₃ is nilpotent, T ₃ has SVEP, hence f ₂ ( z ) = 0, ( T ₁ − z ) f ₁ ( z ) = 0. Since T ₁ is a 2-isometric operator, T ₁ has SVEP by [8 , Theorem 2], then f ₁ ( z ) = 0. Consequently, T has SVEP. Now suppose that T is a polynomially k -quasi-2-isometric operator. Then p ( T ) is a k -quasi-2-isometric operator for some nonconstant complex polynomial p , and hence p ( T ) has SVEP. Therefore, T has SVEP by [13 , Theorem 3.3.9].               ☐ Since the SVEP for T entails that generalized Browder’s theorem holds for T , i.e. σ_BW ( T ) = σ _D ( T ), where σ_D ( T ) denotes the Drazin spectrum, a sufficient condition for an operator T satisfying generalized Browder’s theorem to satisfy generalized Weyl’s theorem is that T is polaroid. In [14] , Patel showed that Weyl’s theorem holds for 2-isometric operator. Then we have the following result: Theorem 2.13. If T is a polynomially k-quasi -2 -isometric operator, then generalized Weyl’s theorem holds for T, so does Weyl’s theorem. Proof. It is obvious from Theorem 2.10, Theorem 2.12 and the statements of the above.               ☐