新算子等式下谱的共同性质

本文研究了无限维Banach空间上满足算子等式组的有界线性算子A和BCk的谱性质,其中k为某个非负整数.具体而言,设A,B,C是定义在无限维Banach空间X上的有界线性算子满足C^(k)BC^(k)=AC^(k)和C^(k)BA^(k)=A^(k+1).本文从正则集的角度证明了算子A和BC^(k)的19类谱是一致的.特别地,我们利用A和BC^(k)的Fredholm谱相等,获得了A和BC^(k)的广义Drazin-Riesz可逆性是等价的.这些结果是对Yan[7]中结论的推广.

Jacobson's Lemma for Generalized Drazin–Riesz Inverses

For bounded linear operators A,B,C and D on a Banach space X,we show that if BAC=BDB and CDB=CAC then I-AC is generalized Drazin-Riesz invertible if and only if I-BD is generalized Drazin-Riesz invertible,which gives a positive answer to Question 4.9 in Yan,Zeng and Zhu[Complex Anal.Oper.Theory 14,Paper No.12(2020)].In particular,we show that Jacobson’s lemma holds for generalized Drazin-Riesz inverses.