Strong Irreducibility Of Operator Weighted Shifts

let H be a complex separable hilbert space and let {W1, W2, W3, ... }be a uniform bounded inverse sequence of operators on H. The operator S on i²(H) = H⊕H ⊕...defined by S:S{x₀, x₁, x₂, ... } = {W₁x₁, W₂x₂, W₃x₃, ...}is called an operator weighted shifts on l'2(H) with the weighted sequence Wk, denoted by S {Wk}_k=1^∞ Then 5 is bounded and ||s|| = sup ||Wk||.If dim H < ∞, S is called finite multiplicity;If dim H = N₀, S is called infinite multiplicity.The operator weighted shifts was first studied by Lamber (1971). The operator class is a natural gerieralzation of the scalar weighted shifts operator, they own many similar properties and have universal application. In the resent years, The operator weighted shifts is the operator class which has been paid attention to.Jue Xian Li [3] gave sufficent and necessary condition that finite multipic-ity operator weighted shift is strongly irreducible in his doctor paper. Thus, it is a ratural problem when an infinite multipilicity operator weighted shifts has strong irreducibility. This paper answers this problem partly.Theorem 2.1 Let S {W_k}_k=1^∞.If that is bounded implies A = λI(orA = λI + Q).λ∈ C, then S∈ (SI).Theorem 2.2 LetS {Wk}_k=1^∞ If that{W_n^-1W_n-1^-1... W_<sup>-1AW₁W₂...Wn}_n=1^∞ is bounded implies A G (SI) ,then S∈ (SI).Theorem 2.3 let 5 {Wk}_k=1^∞ If that{W_n^-1W_n-1^-1... W_<sup>-1AW₁W₂...Wn}_n=1^∞ implies σ(A) is a single point set, then S∈(SI).Except for gaving the above there theorems, in order to explain the sense of the theorems, this paper gave two examples in third section.Example 1Q _<sub>â– \r _<sub>100 â– ??00010 â– ??00001 â– ??00000 ???10000 ???11Let Wi = VUW2 = Vf1, W3 = V2, W4 = V2, \V, = Kf \ Wb = W7 = Vu Ws = VUW* = VUWW = VfWVn = Vf1, Wv> Wvi = W14 = W1S = Wl0 = V2, Wa = Wi& = Wls = W,{) W21 = W22 = W2i = W24 = W20 - V-s W26 = W27 = W28 = Ww = Wsa = Vf1 W31 = W32 = W33 = VK34 = W-Vo - W36 = V1 W37 = Wss = Wi9 - W40 - W41 = W42 = V,-' â–  â–  â– 0Wi00 ???H00w20 ???H000w3 ?â– ?H0000 ???His strongly imlucible.From example 1 we can see the operator weglited shifts satisfied the con-dion of theorem 2.1 and theorem 2.3, and its sense can not be denied meantime, it represents also a class of operator.