| Denote l2 as the Hibert space of all sequences that are square summarizable,that is for any x∈l2, x = (x1, x2,…),sum from n=1 to∞|xn|2<+∞. {ωn}n∞is theabounded sequence of complex number ,and for any n ,ωn≠0, a numerical backward shift operator W in l2 with weight sequence {ωn}n∞acts as follows:Wen =ωnen-1,n = 1,2,…We0 =0defines W* as :W*en=ωnen+1that it is called a numerical forward shift operator with weight sequence {ωn}n∞in l2.Denote W and W* as numerical weighted shifts.Numerical weighted shift operators have a generalization.let Cn = C×…×C is the complex space of rank n,{Wk}k=1∞is the abounded operator sequence in Cn.Denote S inκ+:= (?)k=0∞Cn asS(x0,…,xk,…)=(W1x1,…,Wkxk,…),(?)(xk)∈κ+S is called a backward operator weighted shift with weight sequence{Wk}k=1∞and denote S {Wk}k=1∞.A discrete dynamical system is simply a continuous function f : X→X, where x is the complex separable metric space .For x∈X,the orbit of x under f is defined as orb(f,x) = {x, f(x), f2(x),…}. A point x∈X is a period point for f if fp(x) = x for some p≥1. If there exits a point x∈X such thatX = orb(f, x), then we call (X, f) transitive . Birkhoff's Transitive Theorem A continuous map T of a complete ,separable metric space X is transitive if and only if for every pair U,V of nonempty open subsets of X ,there is a non-negative integern such thatT-n(U)∩V≠(?).(or equivalently U∩T-n(V)≠(?)) .f has sensitive dependence initial conditions if there is a constantδ> 0,such that for any x∈X and any neighborhood U of X , there exits a point y∈X such that d(fn(x), fn(y)) >δ,where d denotes the metric on X .Devaney's Definition of Chaos Suppose f : X→X is a continuous function on a complete separable metric space X ,then f is chaos if :(1) the periodic points for f are dense in X;(2) f is transitive ;(3) f has sensitive dependence on initial condition . where (1)+(2) implies (3).This paper is interested in some dynamical properties about operator weighted shift .As operator weighted shifts ,we firstly introduced some dynamicalproperties about numerical weighted shift .Rolewicz's Theorem For every scalarλof modulus > 1 ,the operatorλB is hypercyclic on lp for each 0 < p <∞, where B is the operator on lp byB(x1, x2, x3,…) = (x2, x3,…)λB is hypercyclic , that is transitive .In particular ,p = 2, we can sayλB is chaotic if its periodic points are dense in l2. That we can also sayλB is chaotic if it has a non-trivial periodic point .Whether the other numerical weighted shift is as the same ? Gross-Erdmann proved it :Gross-Erdmann's Theorem Let T : l2→l2 be a unilateral weighted backward shift with weight sequence (an)n∈N,Then the following assertion are equivalent: (1) T is chaotic ;(2) T is hypercyclic and has a non-trivial period point ;(3) T has a non-trivial periodic point;(4) the series sum from n=1 to∞(?)-1 converge in l2 .In this paper ,we extended the above theorem to operator weighted shifts.Theorem S - {Wk}k=1∞is an operator weighted ,the weighted sequence {Wk)k=1∞is the matrix of rank 2,(1) If W1 = W2 =…= W =(?),then S{Wk}k=1∞is chaotic if and only if |λ1| > 1 and |λ2| > 1.(2)If Wk is written as Wk =(?),α(n)=|multiply from k=1 to nμk|,β(n)=|multiply from k=1 to n vk|.Let a1 =ω1, a2 =μ1ω2+v2a1,…,an=μ1·μ2…μn-1·ωn+vn·an-1,then S is chaotic if and only if the seriesconverge in l2 .Theorem S {Wk}k=1∞is an operator weighted, the weighted sequence {Wk}k=1∞is the matrix of rank n,(1) If Wk is written as W1 = W2 =…= W =(?),then Sis chaotic if and only if |λ|> 1.(2) If Wk is written as W1 = W2 =…= W, where W is the Jordan matrix with l Jordan block, W is written as W = B1 (?) B2(?)…(?)Bl,λi is the eigenvectors of Bi , where i = 1, 2,…, l, then S is chaotic if and only if|λi|>1,i=1,2...,l.
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