--Migrating from MSE since it might fit better here--
Base result
The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as:
$$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \lambda_i(B)$$
where $A, B$ are Hermitian matrices and $\lambda_i(X)$ with $X$ Hermitian denotes the $i$'th eigenvalue of $X$, such that w.l.o.g. $\lambda_1(X) \geq \dots \geq \lambda_n(X).$
Going from Hermitian matrices to kernel operators
Consider the operator induced by a stationary positive semi-definite kernel $k(x, y) := k(\Vert x - y \Vert)$
$$Tf(x) = \int_0^1 k(x,y) f(y) d\mu$$ with $T \in L^2([0, 1], \mu)$ where $\mu$ is some arbitrary measure. I'm interested in Mercer kernels, i.e. kernels which have an eigenfunction expansion of the form
$$k(x, y) := \sum_i \lambda_i \phi_i(x) \phi_i(y)$$
where $\big \langle \phi_i, \phi_j \big\rangle_{L^2(\mu)} = \delta_{ij}$, and $\big\langle k(x, \cdot), \phi_i \big\rangle_{L^2(\mu)} = \lambda_i \phi_i(x)$, and the eigenvalues $\lambda_i$ decay very fast (i.e. polynomial and exponential decay).
Actual Question
I feel that Ky Fan's inequality should apply directly to kernel operators as well, i.e. considering $k_1(\cdot, \cdot), k_2(\cdot, \cdot)$ as before, under what conditions can we say
$$\sum_i \lambda_i(k_1 + k_2) \leq \sum_i \lambda_i(k_1) + \lambda_i(k_2) $$
as in the Hermitian matrix case?
