  In this paper, we report on new results related to the existence of an
adjoint for operators on separable Banach spaces and discuss a few interesting
applications. (Some results are new even for Hilbert spaces.) Our first two
applications provide an extension of the Poincar\'{e} inequality and the
Stone-von Neumann version of the spectral theorem for a large class of
$C_0$-generators of contraction semigroups on separable Banach spaces. Our
third application provides a natural extension of the Schatten-class of
operators to all separable Banach spaces. As a part of this program, we
introduce a new class of separable Banach spaces. As a side benefit, these
spaces also provide a natural framework for the (rigorous) construction of the
path integral as envisioned by Feynman.
