One version of this theorem states that if $E$ is a complex vector space with $\dim(E)<\infty$, and $A$ an unitary sub-algebra of $\mathcal{L}(E)$ for which there are no non-trivial subspaces $F$ invariant by all the elements of $A$ (simultaneously), then $A=\mathcal{L}(E)$. In other words, if $A$ is a strict unitary subalgebra of $\mathcal{L}(E)$, then there is a subspace $F$ non-trivial invariant by all elements of A.
Do you know some simple examples of such $A$ ? Could we prove some results concerning simultaneous triangularization using directly this theorem? I have only found other theorems resulting of this one, or examples not very simple.
Thanks!
Two examples.
i) Two randomly chosen complex $n\times n$ matrices $A,B$ span the full matrix algebra with probability $1$. cf. my post in
Probability that two random matrices span the full matrix algebra
Note that it's not obvious...
ii) -Easier- Let $A,B$ be two $2\times 2$ matrices s.t. $e^Ae^B=e^{A+B}\not= e^Be^A$. Show that $A,B$ are simultaneously triangularizable.