  In a previous paper, the authors introduced the idea of a symmetric pair of
operators as a way to compute self-adjoint extensions of symmetric operators.
In brief, a symmetric pair consists of two densely defined linear operators $A$
and $B$, with $A \subseteq B^\star$ and $B \subseteq A^\star$. In this paper,
we will show by example that symmetric pairs may be used to deduce closability
of operators and sometimes even compute adjoints. In particular, we prove that
the Malliavin derivative and Skorokhod integral of stochastic calculus are
closable, and the closures are mutually adjoint. We also prove that the basic
involutions of Tomita-Takesaki theory are closable and that their closures are
mutually adjoint. Applications to functions of finite energy on infinite graphs
are also discussed, wherein the Laplace operator and inclusion operator form a
symmetric pair.
