On control subspaces of minimal dimension

The quantity "$\operatorname{disc}$" for a (bounded linear) operator was introduced by N. K. Nikol'skii and V. I. Vasjunin, namely,
$$ \operatorname{disc}T=\sup_{E\in\mathcal R(T)}\min\{\dim E'\colon E'\subset E,\ E'\in\mathcal R(T)\}, $$
where $\mathcal R(T)$ is the family of all finite dimensional reproducing subspaces for an operator $T$. We give sufficient conditions on operators $T$ under which $\operatorname{disc}T=\infty$. In particular, we show that there exists an operator $T$ with $\operatorname{disc}T=\infty$ and such that $T$ can be represented in the form $T=T_1\oplus T_2$ with $\operatorname{disc}T_1=\operatorname{disc}T_2=1$.