Let $A:X\to X$ be a compact linear operator on a normed space $X$. Define $L = I - A$. Also suppose that $I-A$ is bijective (although I'm not sure if this is relevant to the question I am going to ask).
Can we say that for some convergent sequence $\{\varphi_n\}$ in $X$ that $$ \lim_{n\to \infty} L {\varphi_n} = L \lim_{n\to \infty} {\varphi_n} $$ If so, why?
For the desired conclusion $$ \text{for every convergent sequence $\varphi_n$ in $X$,}\quad \lim_{n\to \infty} L {\varphi_n} = L \lim_{n\to \infty} {\varphi_n} $$ a mathematician would say: "$L$ is continuous." If your definition of "compact linear operator" includes continuous, then (since $I$ is continuous and the sum of continuous operators is continuous) we may answer YES to the question.