  In this article, we present a solution to the problem: "Which type of linear
operators can be realized by the Dirichlet-to-Neumann operator associated with
the operator $-\Delta-a(z)\frac{\partial^{2}}{\partial z^2}$ on an extension
problem?", which was raised in the pioneering work [Comm. Par.Diff. Equ. 32
(2007)] by Caffarelli and Silvestre. And, we even go beyond by replacing the
negative Laplace operator $-\Delta$ on $\mathbb{R}^{d}$ by an $m$-accretive
operator $A$ on a general Banach space $X$ and the Dirichlet-to-Neumann
operator by the Dirichlet-to-Wentzell operator. We provide a uniquness of the
extension problem in the general Banach spaces framework, which seems to be new
in the literature and independent interest. We develop a type of functional
calculus using probabilistic tools from excursion theory. With our method, we
are able to characterize all linear operators $\psi(A)$, where $\psi$ is a
complete Bernstein function. Our result provides a new characterization of
Phillips' subordination theorem within the class of complete Bernstein
functions.
