Similarity Of Perturbations And The Closures Of The (u+k)-Orbit Of Operator Weighted Shifts

Let H be a complex separable Hilbert space, T be a bounded linear operator on H.The (u+k)-orbit of T is defined as (u+k)(T)={R^-1TR: R invertible of the formunitary plus compact}, the norm closure of the (u+k)-orbit of T is denoted as (?).If {A_k}_k≥0 be a uniformly bounded sequence of Invertible operators on H, H_n=H,(?)=sum from n=0 to +∞(⊕H_n) the unilateral operator weighted shift S on (?) with the weightedsequence {A_k}_k≥0 is defined as S(x₀,x₁,x₂,…)=(0, A₀x₀,A₁x₁,…), (x_n)_n∈(?), denoted byS～{A_k}_k≥0. Let C be the complex plane, H_n=H=Cⁿ=⊕_i=1ⁿC, S is also said to be an-multiple unilateral operator weighted shift.This thesis contains three chapters, we give the preliminaries on this paper in the firstchapter, and discuss the similarity of perturbation of operator weighted shift in the second paper.and describe the closure of the (u+k)-orbit of certain essentially normal operator weighted shiftin the last chapter. We get the following main results:1. Let 1＜dim H=n＜+∞, S～{A_k}_k≥0, P be of a lower triangular operator matrixrepresentation with respect to (?), whose elements satisfy the main diagonal elements andsubdiagonal elements are 0, respectively, If sup_i{‖A_i+1‖_H‖A_i^-1‖_H}≤r＜1 and |‖PS_-1‖|＜1,then S+P is similar to S. where S_-1 is of upper triangular matrix representation whose allelements are 0 except the subdiagonal (A₀^-1, A₁^-1, A₂^-1,…). When T has matrix representationT=[T_ij] with respect to (?), written |‖T‖|=sum from i sum from j(‖T_ij‖_H).2. Let T～{W_k}_k≥1 be an essentially normal operator on (?)=⊕_k=0⁺∞Cⁿ, W_k=(w_ij^(k)_n×n isa upper triangular matrix with w_ii^(k)＞0 and 0＜(?){w_ii⁽ⁿ⁾)}=α_i≤(?){w_ii⁽ⁿ⁾}=β_i＜ (?){w_{i+1 i+1}⁽ⁿ⁾}=α_i+1 for all i=1,2,…, n-1. Then (?)={R is a bounded linearoperatoron (?):R satisfies (1). R is an essentially normal operator; (2).σ(R)={z∈C:|z|≤β_n};(3).σ_e(R)=U_i=1ⁿ{z∈C:α_i≤|z|≤β_i}; (4). ind(R-λ)=-n for all|λ|＜α₁, ind(R-λ)=-(n-i) for allβ_i＜|λ|＜α_i+1, i=1, 2,…,n-1.}.3. Let T～{W_k}_k≥1 satisfy the same conditions of 2, andα_i=β_i=γ_i for i=1,2,…,n-1. Then (?)={R is a bounded linear operator on (?):R satisfies (1). R is anessentially normal operator;(2),σ(R)={z∈C:|z|≤γ_n};(3).σ_e(R)=∪_i=1ⁿ{z∈C:|z|=γ_i},(4). ind(R-λ)=-n for all |λ|＜γ₁, ind(R-λ)=-(n-i) for allγ_i＜|λ|＜γ_i+1,i=1,2,…, n-1.}.