<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Farida Lombarkia</style></author><author><style face="normal" font="default" size="100%">Amina Boussaid</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Operator equations and inner inverses of elementary operators</style></title><secondary-title><style face="normal" font="default" size="100%">Linear and Multilinear AlgebraLinear and Multilinear Algebra</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2019</style></year><pub-dates><date><style  face="normal" font="default" size="100%">2019</style></date></pub-dates></dates><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">Let&amp;nbsp;E,F,G,D&amp;nbsp;be infinite complex Banach spaces and&amp;nbsp;B(F,E)&amp;nbsp;the Banach space of all bounded linear operators from&amp;nbsp;F&amp;nbsp;into&amp;nbsp;E. Consider&amp;nbsp;A1,A2∈B(F,E),&amp;nbsp;B1,B2∈B(D,G)B1,B2∈B(D,G). Let&amp;nbsp;MA1,B1:X→A1XB1&amp;nbsp;be the multiplication operator on&amp;nbsp;B(G,F)&amp;nbsp;induced by&amp;nbsp;A1,B1. In particular,&amp;nbsp;LA1=MA1,I&amp;nbsp;and&amp;nbsp;RB1=MI,B1, where&amp;nbsp;I&amp;nbsp;is the identity operator are the left and the right multiplication operators, respectively. The elementary operator Ψ defined on&amp;nbsp;B(G,F)B(G,F)&amp;nbsp;is the sum of two multiplication operators Ψ=MA1,B1+MA2,B2. This paper gives necessary and sufficient conditions for the existence of a common solution of the operator equations&amp;nbsp;MA1,B1(X)=C1 and&amp;nbsp;MA2,B2(X)=C2 and derive a new representation of the general common solution via the inner inverse of the elementary operator Ψ; we apply this result to determine new necessary and sufficient conditions for the existence of a Hermitian solution and a representation of the general Hermitian solution to the operator equation&amp;nbsp;MA,B(X)=C. As a consequence, we obtain well-known results of Dajic´&amp;nbsp;and Koliha.</style></abstract></record></records></xml>