Let $X$ be a banach space. We say a sequence $(x_n)$ is minimal if for each $k$, $x_k\notin [x_n]_{n\neq k}$, where $[x,y,z,\cdots]$ is the closed linear span of the vectors.
For $T\in\mathcal{L}(X)$, a bounded linear operator on $X$, $Orb(T,x)=\{x,Tx,T^2x,\cdots\}$ is the orbit of $x$ under $T$.
The question I am asking is that when an operator has a vector whose orbit is minimal. Since the existence of such an operator would imply the existence of almost invariant half spaces for a large class of operators, including all quasi nilpotent ones in particular, such a condition would be very interesting.
I am not quite sure where to look. But any conditions on either the operator or the vector would be helpful.