Let $\mathfrak{S}_1$ be the space of trace-class self-adjoint operators on $L^2(\mathbb{R}^n)$, and $\psi\in L^2(\mathbb{R}^n)$ such that $\int |\psi|^2 = 1$. Is there a projection from $\mathfrak{S}_1$ onto $$\left\{ \left|\varphi\right>\left<\psi\right|+\left|\psi\right>\left<\varphi\right|| \varphi\in\{\psi\}^{\perp}_{L^2}\right\} ?$$

If needed, one can change $\mathfrak{S}_1$ for $\mathfrak{S}_1 \cap \{ tr \cdot = 0 \}$ its subspace of vanishing-trace operators. A first question would be whether $\left\{ \left|\varphi\right>\left<\psi\right|+\left|\psi\right>\left<\varphi\right|| \varphi\in\{\psi\}^{\perp}_{L^2}\right\}$ is closed in $\mathfrak{S}_1$.