  Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic
von Neumann algebra with all atoms having the same trace) acting on a (not
necessarily separable) Hilbert space $H$ equipped with a semifinite faithful
normal trace $\tau$. Let $E(\mathcal{M},\tau) $ be a symmetric operator space
affiliated with $ \mathcal{M} $, whose norm is order continuous and is not
proportional to the Hilbertian norm $\left\|\cdot\right\|_2$ on
$L_2(\mathcal{M},\tau)$. We obtain general description of all bounded hermitian
operators on $E(\mathcal{M},\tau)$. This is the first time that the description
of hermitian operators on asymmetric operator space (even for a noncommutative
$L_p$-space) is obtained in the setting of general (non-hyperfinite) von
Neumann algebras. As an application, we resolve a long-standing open problem
concerning the description of isometries raised in the 1980s, which generalizes
and unifies numerous earlier results.
