Abstract
The study of extensions of densely defined operators has led to operators whose image have a real part that is contained wholly in the left hand side of the imaginary line of the complex plane. This is achieved through the fact that when the numerical range of a sesquilinear operator, is not the whole plane, then it is contained in the half plane and is given by
Rerϕx+k0 x2≥ 0 , for all x∈D(ϕ)
. ---------------------------- ------------------------------------------(1)
The discussions led to the function whose image is given be rϕ(x)+k0x2]≥0 leading to sesquilinear function whose associated operator is defined by equation Re Sx,x≤ 0 for all x∈DS, an operator whose real part is less than or equal to zero. This operator is called dissipative operator . In this paper we seek to determine if there can be an extension for our new operator given that its numerical range is a half plane.