My simplified version of the question:
Given a symmetric matrix $A = PDP^T$.
Let $\tilde D$ be the matrix obtained by replacing the non-positive entries in $D$ with zero's.
Then from what I am reading, the notes called $A\tilde D A^{T}$ the projection of $A$ onto its positive eigenspace. Why is this a projection of a matrix?
(See details below.)
Given the Airy operator $H = -\frac{d^2}{dx^2} + x$, it is easy to see that $\text{Ai}(x-\lambda)$ are the generalized eigenfunctions of $H$ w.r.t. eigenvalue $\lambda$ (this follows from that $\text{Ai}(x)$ solves $y''=xy$).
Define the $\text{Airy}_2$ kernel as $$K_{\text{Ai}} (x,y)= \int_0^\infty \text{Ai}(x+z)\text{Ai}(y+z)dz.$$ It is the projection of $H$ onto its negative generalized eigenspace.
Define $Af(x) = \int_{\mathbb{R}} \text{Ai}(x-z)f(z) dz$. The notes justified the above by saying $``K_{\text{Ai}} = APA^\star"$ where $P$ is the multiplication with ${\bf 1}_{x<0}$ operator. I guess here $K_{\text{Ai}}$ is understood as an integral operator rather than a function of two variables.