Dissipative Cayley-transform contraction ($\Vert (I-hL/2{)}^{-1}(I+hL/2){\Vert }_2\le 1$)

If $\mathrm{Re}⟨Lv,v⟩\le 0$ in a finite-dimensional Hilbert space, $I-hL/2$ is invertible for every positive $h$. The relation $v_{+}-v_{-}=(h/2)L(v_{+}+v_{-})$ gives $\Vert v_{+}{\Vert }^2-\Vert v_{-}{\Vert }^2=(h/2)\mathrm{Re}⟨L(v_{+}+v_{-}),v_{+}+v_{-}⟩\le 0$. Thus the Crank--Nicolson method contracts for such a linear system even when $L$ is a non-normal matrix. This is a direct energy result, not an unqualified inequality by a scalar rational function evaluated at a logarithmic norm.