Relatively open operators and the ubiquitous concept

A linear operator $T:D(T)\subset X\rightarrow Y$, when $X$ and $Y$ are normed spaces, is called ubiquitously open (UO) if every infinite dimensional subspace $M$ of $D(T)$ contains another such subspace $N$ for which $T|N$ is open (in the relative sense). The following properties are shown to be equivalent: (i) $T$ is UO, (ii) $T$ is ubiquitously almost open, (iii) no infinite dimensional restriction of $T$ is injective and precompact, (iv) either $T$ is upper semi-Fredholm or $T$ has finite dimensional range, (v) for each infinite dimensional subspace $M$ of $D(T)$, we have $\dim (T|M)^{-1}(0)+\Delta (T|M)>0$. In case $T$ is closed and $X$ and $Y$ are Banach spaces, $T$ is UO if and only if $\overline{TM}\subset T\overline{M}$ for every linear subspace $M$ of $X$.