Solvability of certain quadratic operator equations and representations of Drazin inverses

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• 作者：Qingxiang Xu^a ; ^b ; ¹ ; ^{qingxiang_xu@126.com} ; ^{qxxu@shnu.edu.cn} ; Chuanning Song^a ; ¹ ; ^{songning@shnu.edu.cn} ; ^{songning1962@163.com} ; Li Zhang^a ; ^{287611164@qq.com}
• 关键词：15A09 ; 15A24 ; 47A62
• 刊名：Linear Algebra and its Applications
• 出版年：2013
• 出版时间：15 July, 2013
• 年：2013
• 卷：439
• 期：2
• 页码：291-309
• 全文大小：403 K

文摘

Let E be a Banach space, be the set of bounded linear operators on E, and I be the identity operator on E. In this paper the solvability of the quadratic operator equations and is studied, where is invertible and is nilpotent. It is proved that the first quadratic operator equation is solvable if furthermore or for and . A counterexample is given such that the second quadratic operator equation is unsolvable. As an application, in the general setting of bounded linear operators on Banach spaces, a new expression of the Drazin inverse is given under the condition that are Drazin invertible and .