Consider any two real symmetric matrices $A,B\in\mathbb{S}^n$ with spectral decompositions: $$A=U_A D_A U_A',\quad B=U_BD_BU_B'$$ Now, let $$A^-\equiv U_A\min(D_A,0)U_A',\quad B^-=U_B\min(D_B,0)U_B'$$ where $\min(D,0)$ operates on every entry on the diagonal.
Is there any matrix norm such that $\forall A,B\in\mathbb{S}^n$: $$\|A^--B^-\|\leq\|A-B\|$$