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Let $X$ be an infinite dimensional Banach space (separable if it helps) and $T: X\rightarrow X$ a linear automorphism.

Are there any conditions on $T$ that guarantee that $T$ has simple eigenvalues?

I am aware of one condition that guarantees that the adjoint of $T$ has a simple eigenvalue (or no eigenvalue at all.) Namely that $T$ be supercyclic, or that there exists an $x \in X$ such that the projective orbit is dense, i.e.

$$\overline{ \{\lambda T^n x \ | \ \lambda \in \mathbb{C}, n \in \mathbb{Z} \}} = X$$

I'm not terribly interested in results that apply only for Hilbert spaces.

Thanks in advance!

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