The Extremal Values of $\operatorname{tr}(AX)$ over Symmetric Matrices between $aI$ and $bI$

In the setting of Symmetric and Positive Semidefinite Matrices, the Operator Norm and Spectral Notation, let $n\in\mathbb{N}$, let $a,b\in\mathbb{R}$ with $a\le b$, let

$$\mathcal{A}_{a,b}=\{A\in\mathcal{S}(n):aI_{n}\preceq A\preceq bI_{n}\},$$

and let $X\in\mathcal{S}(n)$. Write $\operatorname{tr}$ for the trace and $\lambda_{1}(X)\ge\dots\ge\lambda_{n}(X)$ for the ordered eigenvalues of $X$.

(i) Maximum. There is $A^{*}\in\mathcal{A}_{a,b}$ such that

$$\operatorname{tr}(AX)\le\operatorname{tr}(A^{*}X)=b\sum_{k\in[n],\ \lambda_{k}(X)>0}\lambda_{k}(X)+a\sum_{k\in[n],\ \lambda_{k}(X)<0}\lambda_{k}(X)\qquad\text{for every }A\in\mathcal{A}_{a,b}.$$

(ii) Minimum. There is $A_{*}\in\mathcal{A}_{a,b}$ such that

$$\operatorname{tr}(AX)\ge\operatorname{tr}(A_{*}X)=a\sum_{k\in[n],\ \lambda_{k}(X)>0}\lambda_{k}(X)+b\sum_{k\in[n],\ \lambda_{k}(X)<0}\lambda_{k}(X)\qquad\text{for every }A\in\mathcal{A}_{a,b}.$$