Eigenfunction of an integral operator

Remarks

So an eigenfunction is an eigenvector of $T_K$ in the vector space $L^2[a, b]$. When $\mu \ne 0$, the eigenfunctions for $\mu$ are the nonzero solutions of the homogeneous equation $\varphi = \lambda T_K \varphi$ with $\lambda = 1/\mu$.

Example: for $K(x, y) = \min(x, y)$ on $[0, 1]$ and $\varphi(x) = \sin(\pi x / 2)$, one has $\int_0^1 \min(x, y) \varphi(y) \, dy = \frac{4}{\pi^2} \varphi(x)$. To check, call the left side $g(x)$; then $g(0) = 0$, $g'(1) = 0$ and $g'' = -\varphi$, and $\frac{4}{\pi^2} \varphi$ satisfies the same three conditions, which determine $g$.

Non-example: for the same kernel, $\varphi = 1$ is not an eigenfunction, since $\int_0^1 \min(x, y) \, dy = x - x^2/2$ is not a constant multiple of $1$.

The same definition is used with $L^2[a, b]$ replaced by $C[a, b]$ for continuous kernels, or by the $L^2$ space of a measure space. For continuous kernels and $\mu \ne 0$, every eigenfunction in $L^2$ agrees almost everywhere with a continuous one.