ON THE DERTERMINATION OF EXTENSIONS OF CLOSABLE OPERATORS AND THE NUMERICAL RANGE

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

Rex+k0 x2≥ 0 , for all x∈D(ϕ)Image removed. . ---------------------------- ------------------------------------------(1)

The discussions led to the function whose image is given be rϕ(x)+k0x2]≥0Image removed. leading to sesquilinear function whose associated operator is defined by equation Re Sx,x≤ 0  for all  x∈DS,Image removed.  an operator whose real part is less than or equal to zero. This operator is called dissipative operatorImage removed.. 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.

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Author
Justus K. Mile