Operators from Banach spaces to complex interpolation spaces

Abstract:

Given a closed linear operator $A$ with dense domain in a Banach space $X$, M. Schechter [4] utilized the Lebesgue integral to construct a family of bounded linear operators from $X$ to the Calderón complex interpolation space ${(X,D(A))_8}$ [2], where $D(A)$, the domain of $A$ in $X$, is a Banach space under the norm \[ ||x|{|_{D(A)}} = ||x|| + ||Ax||.\] In this paper we utilize the complex functional calculus, which provides a more natural setting, to construct a similar family of operators. At the same time we achieve a strengthening of the Schechter result, for in the proof of our theorem we make no use of the adjoint ${A^ \ast }$ of $A$ and consequently do not require the domain of $A$ to be dense in $X$. A completely analogous procedure would permit the removal from the Schechter theorem, referred to above, of the hypothesis that the domain of $A$ is dense in $X$.