Complex interpolation of weighted noncommutative $L_p$-spaces

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 $$L_p(d)=x\in L_0(\mathcal{M}):dx+xd\in L_p(\mathcal{M})and\mathrm{\parallel }x{\mathrm{\parallel }}_{L_p(d)}=\mathrm{\parallel }dx+xd{\mathrm{\parallel }}_p\mathrm{.}$$ We show that for $1\le p_0<p_1\le \text{8}$, $0<\theta <1$ and ${\alpha }_0\ge 0,{\alpha }_1\ge 0$ 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 $\frac{1}{p}=\frac{1-\theta }{p_0}+\frac{\theta }{p_1}$ and $\alpha =(1-\theta ){\alpha }_0+\theta {\alpha }_1$.