关于加权移位的几点注记

Some Notes on Weighted Shifts

设H是复、可分无限维Hilbert空间,B(H)为H上的有界线性算子全体。Z、Z_-^+分别代表整数和非负整数I=Z或Z^+,{e_n}_(neI)为H的一组正规正交基,{α_n}_(neI)为有界正数序列,若定义Te_n=α_e_n(?)1,T为H上的有界线性算子,称T为加权移位。如果I=Z^+,称T为单侧加权移位。如果I=Z,称T为双侧加权移位。关于加权移位的研究可见[1]。L.A.Fialkow[2]。[3]讨论了两个加权移位的拟相似以及加权移位拟相似于似幂零算子的条件,并在[3]中提出了如下未解决的问题:

In this paper, operators quasisimilar to an injective non—invertible weighted shift are considered. It is proved that σ(T)(?)σ(S) if S is qussis imilar to an injective non—invertible weighted shift. Additionally, a necessary and sufficient condition for unilateral weighted shift to be quasisimilar to an analytic Toeplitz operator is given.