# Relatively open operators and the ubiquitous concept.

• Volume: 38, Issue: 1, page 69-79
• ISSN: 0214-1493

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## Abstract

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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) &gt; 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.

## How to cite

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Cross, R. W.. "Relatively open operators and the ubiquitous concept.." Publicacions Matemàtiques 38.1 (1994): 69-79. <http://eudml.org/doc/41204>.

@article{Cross1994,
abstract = {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) &gt; 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.},
author = {Cross, R. W.},
journal = {Publicacions Matemàtiques},
keywords = {Operadores lineales; Espacios normados; Bola unidad; relatively open operators; ubiquitously almost open; semi-Fredholm},
language = {eng},
number = {1},
pages = {69-79},
title = {Relatively open operators and the ubiquitous concept.},
url = {http://eudml.org/doc/41204},
volume = {38},
year = {1994},
}

TY - JOUR
AU - Cross, R. W.
TI - Relatively open operators and the ubiquitous concept.
JO - Publicacions Matemàtiques
PY - 1994
VL - 38
IS - 1
SP - 69
EP - 79
AB - 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) &gt; 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.
LA - eng
KW - Operadores lineales; Espacios normados; Bola unidad; relatively open operators; ubiquitously almost open; semi-Fredholm
UR - http://eudml.org/doc/41204
ER -

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