General exact solutions of certain second-order homogeneous algebraic differential equations. Issue 6 (2nd June 2016)

• Record Type: Journal Article
• Authors: He, Lili; Song, Chuanning; Xu, Qingxiang
• Is Part Of: Linear & multilinear algebra. Volume 64:Issue 6(2016)
• Page Start: 1011
• Page End: 1031
• Publication Date: 2016-06-02
• Subjects: Drazin inverse -- algebraic differential equation -- positive semi-definite operator; 15A09 -- 65L80; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2015.1071314
• Languages: English
• ISSNs: 0308-1087

Abstract

Abstract : The general exact solutions of the second-order homogeneous algebraic differential equation (Section.Display) is studied, where and are three known bounded linear operators on a Banach space . If such that is invertible, an anti-triangular operator matrix is defined. When is Drazin invertible, a necessary and sufficient condition is obtained in terms of and its Drazin inverse under which becomes a solution of (0.1). Specifically, a formula for is derived under the assumptions that is a Hilbert space, and are positive semi-definite operators on with the summation being positive definite.