关于*-n-仿正规算子的一个注记(英文)

A Note on *-n-paranormal Operators

设n为正整数,称T为*-n-仿正规算子,若||T^(1+n)x||^(1/(1+n))≥||T~*x||对H中的每个单位向量x都成立;称T为*-n-仿正规算子,若||T^(1+i)x||^(1/(1+i))≥||T~*x||对H中的每个单位向量x及i≥n都成立.若对任意λ∈C,T-λ都是*-n-仿正规算子,则称T为完全*-n-仿正规算子.若T是*-n-仿正规算子,它的近似点谱和联合近似点谱是相等的.另外证明了若T或者T~*是完全*-n-仿正规算子,则Weyl定理对f(T)成立,其中f∈H(σ(T)),还证明了若T~*是完全*-n-仿正规算子,则α-Weyl定理对.f(T)成立.

Let n be a positive integer. An operator T belongs to class ＊-n-paranormal if ｜｜T^（1＋n）x｜｜（1/（1＋n））≥｜｜T^＊x｜｜ for unit vector x. An operator T ∈ B（H） is said to be ＊-n-paranormal if ｜｜T^（1＋i）x｜｜^（1/（1＋i））≥｜｜T^＊x｜｜ for unit vector x and i 〉 n. An operator T ∈ B（H） is said to be totally ＊-n-paranormal, if T - λ is ＊-n-paranormal for every λ∈C. It is showed that if T belongs to class ＊-n-paranormal operators, then its approximate point spectrum and joint approximate point spectrum are identical. We also prove that if either T or T^＊ is totally ＊-n-paranormal, then Weyl＇s theorem holds for f（T） for every f f∈H（σ（T））, and also α-Weyl＇s theorem holds for f（T） if T^＊ is totally ＊-n-paranormal.