关于广义Aluthge变换的谱性质的研究(英文)

Spectral Properties of the Generalized Aluthge Transform

设T∈■(■),T=UT是算子T的极分解,则定义■λ=|T||λU||T|1-λ和■λ(*)=|T*||λU||T*|1-λ,(其中0<λ<1)分别为算子的广义Aluthge变换和广义*-Aluthge变换.本文中主要研究了三者之间的几种谱的关系.同时,还证明了算子T满足修正的Weyl定理当且仅当■λ满足修正的Weyl定理当且仅当■λ(*)满足修正的Weyl定理.最后证明了算子T满足a-Weyl定理当且仅当■λ满足a-Weyl定理.

Let T be a bounded linear operator on a complex Hilbert space H with a polar decomposition T = U ITI. arbitary λ∈ （0,1）, we define the generalized Aluthge transform and the genralized ＊ -Aluthge transform of T by Tλ = ｜T ｜λ^U ｜T ｜^1-λ and T^λ（＊） = ｜T^＊ ｜^λU ｜T^＊｜^1-λ respectively. In this paper, we consider some spectral properties of those operators. Furthermore, it is proved that T satisfies revised Weyl＇s theorem if and only if T^λ （resp.T^λ（＊） does. Meanwhile, we show that a-Weylts theorem holds for T if and only if a-Weyl＇s theorem holds for T^λ,arbitary λ∈ （0,1）.