Let $M\in \mathcal{B}(F)^+$ and $S_1,S_2\in \mathcal{B}(F)$.

I claim that if $S_1$ and $S_2$ are $M$-self adjoint (i.e. $MS_1=S_1^*M$ and $MS_2=S_2^*M$) such that $S_1S_2=S_2S_1$, then $\exists\,(X,\mu)$, $\varphi_1,\varphi_2\in L^\infty(\mu)$ and a unitary operator $U:F\longrightarrow L^2(\mu)$, such that $$U(MS_k)U^*h=\varphi_kh,\;\forall h\in F,\,k=1,2.$$ Do you think that this claim is true?

Thanks.