On bounded operator equation

In this paper, we give the general solutions of bounded operator equation


INTRODUCTION
In 2007  in this section , we give the general solution for the operator equation : , are known and X is unknown operator on H that must be determined . The following theorem introduce the general solution for the nonlinear operator equation (1) is a skew-adjoint operator. Proof: Let X be any solution of equation (1), then Therefore; Z is a skew-adjoint operator Then any solution of operator equation (1) has the form is a general solution of operator equation (1) then its satisfy this equation, thus we get: is a skewadjoint operator. Now, we give the general solution of nonlinear operator equation (1), when B is invertible operator and A is noninvertible operator be an operators and A has closed range. Then equation (1) has solution if and only if

Proof:
In first we reduced equation (1) in to is a general solution of equation (1). To do this substitute in the left side of operator equation And by using the condition * C C  one can have :

Proof:
In first we reduced equation (1) in to is a solution of equation (1). To do this substitute in the left side of operator equation And by using the condition * C C  one can have:   (1) has general solution if and only if

Proof:
In first we reduced equation (1) And by using the condition * C C  one can have:

On bounded operator equation
And by using the condition , one can have:

But if H is a Real Hilbert space then A  is a linear map
Now we give more properties of this map by the following proposition.

On bounded operator equation
, Therefore the map A  is bounded And the following proposition shows in general the map is not necessary one to one Now the following theorem study some properties of the Rang of the map Recall that a mapping f from a ring R into it self is called derivation if . the following remark shows the mapping A  is not derivation .

Remark (2.5): Since
, to illustrate this consider the following example .