Abstract
Our main result asserts that, under some assumptions, the uniformly-to-order continuity of an order bounded orthogonally additive operator between vector lattices together with its horizontally-to-order continuity implies its order continuity (we say that a mapping f : E → F between vector lattices E and F is horizontally-to-order continuous provided f sends laterally increasing order convergent nets in E to order convergent nets in F, and f is uniformly-to-order continuous provided f sends uniformly convergent nets to order convergent nets).
Highlights
Äîñëiäæåííÿìè âçà1ìîçâ'ÿçêiâ ìiæ íàðiçíîþ òà ñóêóïíîþ íåïåðåðâíiñòþ ôóíêöié äâîõ çìiííèõ ìiæ òîïîëîãi÷íèìè ïðîñòîðàìè ìàòåìàòèêè çàéìàþòüñÿ, ïî÷èíàþ÷è ç äèñåðòàöi Ð.
Çà ÿêèõ óìîâ íà âåêòîðíó ðàòêó E ç ãîëîâíîþ ïðîåêòèâíîþ âëàñòèâiñòþ, ïîðÿäêîâî ïîâíó âåêòîðíó ðàòêó F òà ðåãóëÿðíèé îðòîãîíàëüíî àäèòèâíèé îïåðàòîð T ç ãîðèçîíòàëüíî-ïîðÿäêîâîíåïåðåðâíîñòi òà ðiâíîìiðíî-ïîðÿäêîâîíåïåðåðâíîñòi îïåðàòîðà T âèïëèâà1 éîãî ïîðÿäêîâà íåïåðåðâíiñòü?
ñèëüíî ïîðÿäêîâî çáiæíîþ äî ãðàíèöi x ∈ E, ÿêùî iñíó1 ñiòêà (yα)α∈A â E òàêà, ùî yα ↓ 0 òà |xα − x| ≤ yα äëÿ äåÿêîãî α0 ∈ A òà âñiõ α ≥ α0 (ïèøóòü xα −s−→o x);
Summary
Äîñëiäæåííÿìè âçà1ìîçâ'ÿçêiâ ìiæ íàðiçíîþ òà ñóêóïíîþ íåïåðåðâíiñòþ ôóíêöié äâîõ çìiííèõ ìiæ òîïîëîãi÷íèìè ïðîñòîðàìè ìàòåìàòèêè çàéìàþòüñÿ, ïî÷èíàþ÷è ç äèñåðòàöi Ð. Çà ÿêèõ óìîâ íà âåêòîðíó ðàòêó E ç ãîëîâíîþ ïðîåêòèâíîþ âëàñòèâiñòþ, ïîðÿäêîâî ïîâíó âåêòîðíó ðàòêó F òà ðåãóëÿðíèé îðòîãîíàëüíî àäèòèâíèé îïåðàòîð T ç ãîðèçîíòàëüíî-ïîðÿäêîâîíåïåðåðâíîñòi òà ðiâíîìiðíî-ïîðÿäêîâîíåïåðåðâíîñòi îïåðàòîðà T âèïëèâà1 éîãî ïîðÿäêîâà íåïåðåðâíiñòü?
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