Null space Theorem: Let $T: X \rightarrow X\ $ be a compact linear operator on a normed space $X$. Then for every $\lambda \neq 0$ the null space $N(T_\lambda)$, with $T_\lambda = T -\lambda I \ $ is finite dimensional.
Show by simple examples that in Null space Theorem the conditions that $T$ be compact and $\lambda \neq 0 \ $ cannot be omitted.
I tried to use the fact that if $X$ is infinite in dimension then the identity operator cannot be compact, however, I couldn't guarantee the return.
The identity $I:X\to X$ is compact if, and only if, the unit closed ball $\bar B_X$ is compact if, and only if, $X$ is finite dimensional.
omitting the compacity hypothesis : $T = \lambda I$ has $\ker(T-\lambda I) = X$, which can be infinite dimensional.
omitting $\lambda = 0$ : the zero operator is always compact, and its kernel is $X$, which can be infinite dimensional.