Cauchy-Schwarz inequality for positive-semidefinite kernels ($|k(x,y){|}^2\le k(x,x)k(y,y)$)

A real positive-semidefinite kernel satisfies $|k(x,y){|}^2\le k(x,x)k(y,y)$. This follows from positivity of its two-point kernel matrix. If a diagonal value is zero, the corresponding off-diagonal values must also vanish.