Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 8, 383 - 388 Conditions for Hypercyclicity Criterion of a Tuple Bahmann Yousefi and Fariba Ershad Department of Mathematics, Payame Noor University P.O. Box: 71955-1368, Shiraz, Iran b − yousefi@pnu.ac.ir, fershad@pnu.ac.ir Abstract In this paper we give necessary and sufficient conditions for a tuple satisfies the Hypercyclicity Criterion. Mathematics Subject Classification: 47B37; 47B33 Keywords: tuple of operators, hypercyclic vector, Hypercyclicity Crite- rion 1 Introduction Let T =(T 1 ,T 2 ) be a pair of commuting continuous linear operators acting on an infinite dimensional Banach space X . We will let F = {T 1 k 1 T 2 k 2 : k i ≥ 0,i =1, 2}. For x ∈ X , the orbit of x under the tuple T is the set Orb(T ,x)= {Sx : S ∈F}. A vector x is called a hypercyclic vector for T if Orb(T ,x) is dense in X and in this case the tuple T is called hypercyclic. Also, by T d we will refer to the set T d = {S ⊕ S : S ∈F}. We say that T d is hypercyclic provided there exist x 1 ,x 2 ∈ X such that {W (x 1 ⊕ x 2 ): W ∈T d } is dense in X ⊕ X .