Irreducibility And Classification Of Operator Weighted Shifts

Let H be a complex separable Hilbert space and let {W1,W2,...} be auniformly bounded sequence of operators on H. The operator S on 2(H) =H ⊕ H ⊕ ··· de?ned by S : {x0,x1,...} → {0,W1x0,W2x1,...}is called an operator weighted shift on 2(H) with the weight sequence {Wk},denoted by S ～ {Wk}. Then S is bounded and S = sup Wk . The operator kweighted shift was ?rst studied by Lambert. It is a natural generalization ofthe scalar weighted shift operator and owns many properties similar to thoseof the scalar one. In this thesis, operator weighted shifts with one-to-one anddense-range operators as weights are main objects which are studied. This thesis consists of four chapters. In the ?rst chapter, the basic concepts, the background of problems, andthe work ?nished in past are introduced. In the second chapter, we proved that every operator weighted shift S ～{Wk} with one-to-one and dense-range operators {Wk} as weights is similar toan operator weighted shift T ～ {Vk} whose weights {Vk} are one-to-one andpositive operators. Moreover, the weight Vk is unitarily equivalent to |Wk| foreach ?xed k. In the third chapter, we proved that every operator weighted shift S ～ {Wk}, –29 –英 文 摘 要whose weight sequence {Wk} consists of one-to-one and dense-range opera-tors, is similar to an irreducible operator. In addition, we show a class of exam-ples that reducible operators are similar to irreducible ones. In these examples,compact operators and Cp (1 ≤ p < ∞) ones are also contained. All theseexamples show that irreducibility is not an invariant under similar transforms. In the ?nal chapter, the suf?cient and necessary conditions of Cαβ classi?-cation of any multiple contracted operator weighted shifts are given and proved.Especially, in the case of ?nite multiple, some of these conditions can be sim-pli?ed. Finally, two examples are provided to show why these conditions cannot be simpli?ed in general cases.