  Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a
semifinite normal faithful trace $\tau$. Let $d$ be an injective positive
measurable operator with respect to $(\mathcal{M}, \tau)$ such that $d^{-1}$ is
also measurable. Define
  $$L_p(d)={x\in L_0(\mathcal{M}) : dx+xd\in
  L_p(\mathcal{M})}\quad{and}\quad
  \|x\|_{L_p(d)}=\|dx+xd\|_p .$$ We show that for $1\le p_0<p_1\le\8$,
$0<\theta<1$ and $\alpha_0\ge0, \alpha_1\ge0$ the interpolation equality
  $$(L_{p_0}(d^{\alpha_0}), L_{p_1}(d^{\alpha_1}))_\theta
  =L_{p}(d^{\alpha})$$ holds with equivalent norms, where
$\frac1p=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}$ and
$\alpha=(1-\theta)\alpha_0+\theta\alpha_1$.
