This question may be trivial. Consider two positive operators $A,B$ in a $C^*$ Algebra.
Is it possible that the lower bound of the spectrum decreases when they are summed? $$\inf \sigma(A+B) < \inf \sigma(A) + \inf \sigma(B)$$
If all terms are non zero this implies $A, B, A+B$ are invertible and $\|(A+B)^{-1}\|>\|A^{-1}\|+\|B^{-1}\|$ (and would be implied by that inequality in that case).
No, it is not possible. Represent your $C^*$-algebra $\mathcal A$ as bounded linear operators on Hilbert space $\mathcal H$. Then $\inf \sigma(A) = \inf_{\|x\|=1} \langle x, A x \rangle$. Now $$\inf \sigma(A+B) = \inf_{\|x\|=1} \left(\langle x, A x\rangle + \langle x, B x \rangle\right) \ge \inf_{\|x\|=1} \langle x, A x\rangle + \inf_{\|x|=1} \langle x, B x \rangle = \inf \sigma(A) + \inf \sigma(B) $$