Is there a injective compact operator $T$ such that #$sp(T) < \infty$?

Question

Let $X$ a Banach spaces such that $\dim X = \infty$, and $T: X\to X$ a injective compact operator. Is there $T$ such that #$sp(T) < \infty$ ?

Thanks

Answer 1

Yes. There are injective compact quasinilpotent operators (having spectrum $\{0\}$). For example, on a separable infinite dimensional Hilbert space with orthonormal basis $\{e_0,e_1,\ldots\}$, let $T$ be the continuous linear extension of $e_n\mapsto \frac{1}{n+1}e_{n+1}$. Then $T$ is compact and injective, and the spectrum of $T$ is $\{0\}$.