# Does there exist a topologically transitive linear automorphism $l^{\infty}$?

A linear automorphism $$T: X \rightarrow X$$, where $$X$$ is Banach, is said to be topologically transitive if for all $$U$$ and $$V$$ open, there exists $$n \in \mathbb{Z}$$ such that $$T^n(U)\cap V \neq \varnothing.$$

When $$X$$ is separable, transitive is equivalent to $$T$$ having a dense orbit, by a well known theorem of Birkhoff.

Every separable Banach space supports a transitive operator. Concrete examples include certain weighted shifts on $$l^p$$, $$p \in (1, \infty)$$.

Clearly $$T$$ cannot have a dense orbit in $$l^{\infty}$$, since it is not separable.