Given two positive semi-definite matrices $A$ and $B$ (i.e., $A\geq0, B\geq0$). Will the matrix $AB+BA+ABBA$ is positive semi-definite?
I know $AB+BA$ is not necessarily positive semi-definite because the minimum eigenvalue $\lambda_{n}(AB+BA)$ may be negative (see one example in P33 in Page446 in Horn's book "Matrix Analysis"). But $ABBA$ is positive semi-definite. Will the summation of $AB+BA$ and $ABBA$ be positive semi-definite?
Many thanks.