Let $\left(T_{n}\right)_{n}$ be a sequence of operator in a infinitdimensional Hilberspace $H$, defined as restrictions of an operator $T:H\rightarrow H$, on smaller and smaller subsets, by the algorithm in this question (were also additional information about $T$ is provided). There it was shown, that if this sequence is finite, $T$ must have finite rank.
My question is: Is the number $n$, for which the algorithm described here stops, always $\text{rank}T+1$ ? How can we prove that ?
This guess came from the fact, that if $H$ were finite dimensional, it is not too hard to show, that this sequence stops after exactly $\text{rank}T+1$ steps.