Direct finiteness of linear operator ring in infinitely many dimensions

Question

For a vector space of finite dimension, the ring of linear operators End(V) is directly finite i.e. $AB = 1$ implies $BA=1$ for linear operators $A$ and $B$. Is this also true in infinitely many dimensions? If not, what is a counterexample for this? Or is it known to be true in some specific cases, for example in the context of Hilbert spaces?

Answer 1

No, in fact the endomorphism ring of an infinite dimensional vector space is probably the simplest and most commonly given example of a ring that isn't directly finite (a.k.a. Dedekind-finite.)

See for example this which describes the obvious "shift" transformations that make up a pair $a,b$ such that $ab=1$ but $ba\neq 1$.

In fact, the DaRT query for non directly-finite rings only has two positive hits at the moment. The other one is far from trivial to describe.