O a problm of J. d Graaf coctd with algbras of uboudd oprators d Bruij, N.G. Publishd: 01/01/1984 Documt Vrsio Publishr s PDF, also kow as Vrsio of Rcord (icluds fial pag, issu ad volum umbrs) Plas chck th documt vrsio of this publicatio: A submittd mauscript is th author's vrsio of th articl upo submissio ad bfor pr-rviw. Thr ca b importat diffrcs btw th submittd vrsio ad th official publishd vrsio of rcord. Popl itrstd i th rsarch ar advisd to cotact th author for th fial vrsio of th publicatio, or visit th DOI to th publishr's wbsit. Th fial author vrsio ad th gally proof ar vrsios of th publicatio aftr pr rviw. Th fial publishd vrsio faturs th fial layout of th papr icludig th volum, issu ad pag umbrs. Lik to publicatio Citatio for publishd vrsio (APA): Bruij, d, N. G. (1984). O a problm of J. d Graaf coctd with algbras of uboudd oprators. (Eidhov Uivrsity of Tchology : Dpt of Mathmatics : mmoradum; Vol. 8402). Eidhov: Tchisch Hogschool Eidhov. Gral rights Copyright ad moral rights for th publicatios mad accssibl i th public portal ar rtaid by th authors ad/or othr copyright owrs ad it is a coditio of accssig publicatios that usrs rcogis ad abid by th lgal rquirmts associatd with ths rights. Usrs may dowload ad prit o copy of ay publicatio from th public portal for th purpos of privat study or rsarch. You may ot furthr distribut th matrial or us it for ay profit-makig activity or commrcial gai You may frly distribut th URL idtifyig th publicatio i th public portal? Tak dow policy If you bliv that this documt brachs copyright plas cotact us providig dtails, ad w will rmov accss to th work immdiatly ad ivstigat your claim. Dowload dat: 28. Nov. 2017
N 62 TECHNISCHE HOGESCHOOL EINDHOVEN O a problm of J. d Graaf coctd with algbras of uboudd oprators. by N.G. d Bruij Mmoradum 84-02 Fbruary 1984 Dpartmt of Mathmatics ad Computig Scic, Eidhov Uivrsity of Tchology, PO BOX 513, 5600 MB Eidhov, Th Nthrlads.
O a problm of J. d Graaf, coctd with algbras of uboudd oprators by N.G. d Bruij I coctio with his work o uboudd oprators my collagu J. d Graaf raisd th followig qustio. Tak N = {I,2,.. }, ad assum x ~ 0 (m, N), V E > o "10 > 0 E m, N x -E-om < 00 (1) x mil +E-om = co (2) Do thr xist ral umbrs S (m, E N) such that S ~ 0 (m, N) ; "IE> 0 3 0 > 0 ( m, E N -E+om < (0) (3) L m, N 00? (4) Th aswr 1S gativ. W ca giv th followig coutrxampl: 2 x = xp (- (-) ). +m (5) W shall show (1) ad (2), ad that for vry doubl squc {S} with S ~ 0 such that (3) is satisfid, w also hav
2. Sic a ~ 1~ (1) is trivial. W xt show (2). Tak ay E > O. If w succd i fidig 0 such that 0 > 0 ad 3 2 - /(+m) + E - om > 0 (6 ) for ifiitly may pairs (m,), th (2) is obvious. 1 3/2 W tak =(~) Thr ar ifiitly may pairs (m,) with m, ~ ad OE}-! -1 < ~ < (!E)-! (7) (it is irrlvat whthr th lft-had sid is or is ot positiv). If m ad satisfy (7) w hav (~)2 <.f. +m 2 ad (6) follows. W ow tak ay doubl squc {13 } of o-gativ umbrs such that (3) holds. W prov (4) by showig that both /. ad m, E E al3 < 00 ~, / (+m) < P E a 13 < 00, IIlIl m, E ~, / (m+) 2: P (8) (9) whr p has b chos as follows. By (3), with ; Y > 0 such that B 'to.t m, E ~~ -+ym < 00, = 1, thr xists ( 10) W fix p by y p =! l+y ( 11) I th squl w shall us th abbrviatio +m =, so m = 1. For th pairs (m,) occurrig l. (8) w hav 0 < <! II y, ad thrfor 1 y(g - 1),
3. whc! y (+m) < - + ym. Sic Cl S 1 ' w ow hav by (10) : m, h(+mj a < 00. E: N, l (+m) < E W ow gt (8), sic r!y(+m) covrgs. W fially prov (9). W tak ay E with 0 < E < p2, ad w put p(p2_e)=q. By (3) w hav m, E::N!1I1'l - < 00 For th pairs (m,) occurrig ~ (9 ) w hav p S 0 S 1, ad thrfor S _ 0 2 S S. - 9 2 2 2 = p - E - 0 S - E. 0 P It follows that q(m+) 2 (+m)!il - E', whc m, E: q(m+) -E Cl i3 S m < 00. m N, l (+m) ~ p m, E: N Sic -q(m+) r covrgs, this provs (9).