The following optimization is convex: \begin{align} \max_{X\succ0,Y} &\ \ \log \det (I + X +Y + Y^T)\\ & \text{s.t.} \begin{pmatrix} X& Y \\ Y^T & Z \end{pmatrix}\succeq 0, \ \ \ \mathbf{Tr}(X)\le P \end{align} where $Z\succ0$ and $P>0$ are given.
My question is whether the decision variable $Y$ can be transformed or replaced with a be positive semi-definite decision variable. My motivation is to write the optimization as a standard max-det problem or in a nicer form in order to show that the maximizer is unique.