Courant–Fischer characterization of characteristic values

Remarks

Let $\varphi_1, \varphi_2, \dots$ be orthonormal eigenfunctions with $T_K \varphi_j = \varphi_j / \lambda_j$, from the spectral theorem for compact self-adjoint operators. On the span of $\varphi_1, \dots, \varphi_k$, the quotient is at least $1/\lambda_k$; so the max-min is at least $1/\lambda_k$. Any $k$-dimensional $V$ contains a nonzero $\varphi$ orthogonal to $\varphi_1, \dots, \varphi_{k-1}$, and on that orthogonal complement $R_K \le 1/\lambda_k$; so the max-min is exactly $1/\lambda_k$. The min-max form is proved in the same way, with the choice $\psi_j = \varphi_j$ attaining the minimum.

Example: for $K(x, y) = \min(x, y)$ on $[0, 1]$, $\lambda_k = (k - \tfrac12)^2 \pi^2$. For $k = 1$ the theorem says $\int_0^1 \int_0^1 \min(x, y) \varphi(x) \varphi(y) \, dx \, dy \le \frac{4}{\pi^2} \int_0^1 \varphi^2$ for real $\varphi$, with equality for $\varphi(x) = \sin(\pi x / 2)$.

The symmetry of $K$ is needed: without it the quotient is not real, and the characteristic values need not be real or exist at all. The negative characteristic values satisfy the same formulas applied to the kernel $-K$.

Going deeper

For a student

This is the integral form of the Courant–Fischer min-max theorem for Hermitian matrices. It allows one to compare characteristic values of two kernels without computing them: if $R_{K_1} \le R_{K_2}$ everywhere, then the $k$-th positive characteristic value of $K_1$ is at least that of $K_2$. It is the basis of the Rayleigh–Ritz method, which gives bounds for characteristic values by restricting the quotient to finite-dimensional trial spaces.