Does every invertible $\mathbb{C}$-linear operator have an eigenvalue?

Question

Let $T:X\rightarrow X$ be an invertible linear operator over a complex vector space $X$ (possibly infinite-dimensional), then does $T$ always have an eigenvalue? We may assume that $X$ is a separable Hilbert space if necessary.

I know this is true for finite dimensions by the fundamental theorem of algebra, but how about for infinite dimensions?

Answer 1

Take any bounded operator $S$ with no eigenvalue and choose $N$ such that $\|S\| <N$. Let $T=I+\frac S N$. Then $T$ is invertible but it has no eigenvalue.