Isometries on noncommutative symmetric spaces

Let $\mathscr{M}$ be an atomless semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$ (or else, an atomic von Neumann algebra with all atoms having the same trace) acting on a separable Hilbert space $\mathscr{H}$. Let $E(\mathscr{M},\tau)$ be a separable symmetric space of $\tau$-measurable operators, whose norm is not proportional to the Hilbert norm $\|\cdot\|_2$ on $L_2(\mathscr{M},\tau)$. We provide a description of all bounded Hermitian operators on $E(\mathscr{M},\tau)$ and all surjective linear isometries of this space.