A linear operator T: D(T) ⊂ X → 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) + Δ(T|M) > 0. In case T is closed and X and Y are Banach spaces, T is UO if and only if TM ⊂ TM for every linear subspace M of X.
@article{urn:eudml:doc:41204,
title = {Relatively open operators and the ubiquitous concept.},
journal = {Publicacions Matem\`atiques},
volume = {38},
year = {1994},
pages = {69-79},
mrnumber = {MR1291954},
zbl = {0818.47003},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41204}
}
Cross, R. W. Relatively open operators and the ubiquitous concept.. Publicacions Matemàtiques, Tome 38 (1994) pp. 69-79. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41204/