A question on a trace inequality

Question

Let $A, B\in M_{n}(\mathbb{R})$ be symmetric positive definite matrices. It is easy to see $Tr(A^2+AB^2A)=Tr(A^2+BA^2B)$. Numerical experiments indicate $$Tr[(A^2+AB^2A)^{-1}]\ge Tr[(A^2+BA^2B)^{-1}],~~(1)$$ but it seems difficult to show it.

Remark. When $n=2,3$, by direct computation, (1) is true. Here is an expriment done by matlab:

for s=1:1000

as=randn(4,4);
bs=randn(4,4);
ts=as*as';
rs=bs*bs';
ls=trace(inv(ts^2+ts*rs^2*ts)-inv(ts^2+rs*ts^2*rs))

end

{\bf Updated.} What about $A, B\in M_{n}(\mathbb{C})$ be positive definite Hermitian matrices.

Answer 1

Note first that $A^2+AB^2A=(A+iAB)(A-iBA)$. The reverse product is $(A-iBA)(A+iAB)=A^2+BA^2B-i(BA^2-A^2B)=X-iC$. Thus, the quantity on the left is $\operatorname{Tr} (X-iC)^{-1}$ and that on the right is $\operatorname{Tr} X^{-1}$. Moreover, the self-adjoint complex matrix $X-iC$ is positive definite (as the product of an invertible operator and its adjoint). Similarly, considering the factorization $A^2+AB^2A=(A-iAB)(A+iBA)$, we can write the quantity on the left as $\operatorname{Tr} (X+iC)^{-1}$. Symmetrizing, we see that it will suffice to show that $(X-iC)^{-1}+(X+iC)^{-1}\ge 2X^{-1}$ in the sense of quadratic forms (then the inequality for traces will hold too). We can multiply by $X^{1/2}$ from both sides to reduce it to $(I-iD)^{-1}+(I+iD)^{-1}\ge 2I$ where $D=X^{-1/2}CX^{-1/2}$ and both operators on the left are positive definite. Diagonalizing the self-adjoint operator $iD$, we see that the inequality reduces to $(1+p)^{-1}+(1-p)^{-1}\ge 2$ for $p\in(-1,1)$.

Answer 2

For what it is worth, a weaker conjecture is proved below.

Applying the formula for the derivative of the inverse $$d(M^{-1}) = -M^{-1}\ dM\ M^{-1},$$ to compute the t=0 derivative of the LHS of $$Tr(A^2+A(t^{1/2}B)^2A)^{-1}-Tr(A^2+(t^{1/2}B)A^2(t^{1/2}B))^{-1} \ge 0$$ gives $$Tr(A^{-2}BA^2BA^{-2})\ge Tr(A^{-1}B^2A^{-1})=Tr(BA^{-2}B).$$ Replacing $A^{-2}$ by $P$ gives the weaker conjecture that $$Tr(PBP^{-1}BP)\ge Tr(BPB)$$ for positive B and P.

PROOF OF WEAKER CONJECTURE: By the spectral theorem, we may take $P=\operatorname{Diag}(p_1,p_2,\dotsc)$. Then $$Tr(BPB)=\sum p_j |B_{ij}|^2=\sum |B_{ij}|^2 (p_i+p_j)/2 $$ and $$Tr(PBP^{-1}BP)=\sum |B_{ij}|^2 p_i^2 p_j^{-1}=\sum |B_{ij}|^2 (p_i^2 p_j^{-1}+p_i^{-1}p_j^2)/2.$$ It remains to show that $$p_i^2 p_j^{-1}+p_i^{-1}p_j^2\ge p_i+p_j$$ for positive $p_i, p_j$. By homogeneity we may take $p_i=1$. Multiplying through by $p_j$, the inequality now follows from the identity $$1+p^3-p-p^2=(p-1)^2(1+p)\ge 0.$$ $\square$