Results of semigroup of linear operators in extrapolation spaces

Abstract

Results of an omega-order preserving partial contraction mapping (omega-OCP_n) in generalized spaces are presented in this study. Assumed to be a closed linear operator on a Banach space X with a non-empty resolvent set rho(A) is A in omega-OCP_n. If A is densely defined, the extrapolation spaces X^-1 and X_-1 will be associated with A in agreement. However, X_-1 is a proper closed subspace of X^-1 if A is not densely defined. Then, we demonstrated that the reason these spaces exist is because (X*)^-1 and D(A₀) are naturally isomorphic to (X*)_-1 and (X*)^-1, respectively.