Let $X$ be a Banach space, $A : X \to X$ is a semi-Fredholm operator (not necessarily continuous) with a finite-dimensional kernel, $C : X \to X$ compact such that $\operatorname{Dom}(C)$ contains $\operatorname{Dom}(A)$. Then it is a well-known fact that $A+C$ is also semi-Fredholm with a finite-dimensional kernel.
Obviously, if $S : X \to X$ is any continuous operator with $\operatorname{Dom}(S) = X$, then $A + SC$ is also semi-Fredholm with a finite-dimensional kernel. Moreover, for every positive real number $N$, there exists an integer number $K$ such that $\dim(\ker(A + SC)) \leq K$ for all $S$ which norm does not exceed $N$ (https://arxiv.org/pdf/0902.3045).
Now, the question: How to calculate the function that limits the kernel dimensions of $A + SC$ based on the norm of $S$ in at least some specific spaces?
For example, when $A$ is an identity in $C(\Omega)$ where $\Omega$ is a compact topological space (say $\Omega = [0, 1]$), $C$ is an integral operator with a continuous kernel $c(x,y)$, how to calculate $K$ from $N$ based on $c(x,y)$?