Zero is an eigenvalue of a compact operator on Non-Seperable spaces

Question

Let $X$ be a non seperable Banach Space. If $T$ is a compact operator on $X$, is it true that $Ker(T)\neq \{0\}$, i.e., $0$ is an eigenvalue of $T$.

This question was from my exercise, where $X$ was given to be Hilbert. I wanted to ask if it was possible to prove this in general Banach spaces.

Answer 1

Let $X=\ell^{\infty}$, let $c_n = 2^{-n}$ for every ${n\in\mathbb{N}}$, and let $T:\ell^{\infty}\to\ell^{\infty}$ be defined by $$Tx = (c_nx_n)_{n\in\mathbb{N}}\hspace{1cm} \forall x=(x_n)_{n\in\mathbb{N}}\in\ell^{\infty}.$$ $T$ is compact and $\ker{T}=\{0\}$.