On bounded operator equation

Abstract

College of Education/ Department of mathematicsABSTRACTIn this paper, we give the general solutions of bounded operator equation (1) , where and are noninvertible operator .on complex Hilbert space and properties of the mapping INTRODUCTIONIn 2007 D. S. Djordjevic [2] , find the explicit solution of operator equation for linear operators on Hilbert spaces. Dragana S. in 2008 generalized the result of D. S. Djordjevic in to operator equation [1] , and study solvability of this operator equation .The purpose of this paper is modify the operator equation appear in [1] and give the necessary and sufficient conditions to get the general explicit solution of bounded operator equation where and are noninvertible operator , as well as studied some properties of nonlinear operator mappings , . Let be arbitrary complex Hilbert space , be the space of all bounded linear operators from into . Let , be the mapping defined as follow , ,where and is known operators in .but is unknown operator must be determine and then Rang( )={ } . Also, here we need recall some basic concept of operator that the adjoint operator of is the operator such that , where for all and , an operator is said to be self-adjoint if , and skew-adjoint if ,[4], The moor-penrose inverse of is defined as the operator satisfying the equations , also for the mapping if , for all , then the mapping is derivation .