State diagram for operators with null space or conull space in an ideal of Banach spaces

Pub . Ma . UAB Vol . 30 Nó 1 Maig 1986 STATE DIAGRAM FOR OPERATORS 6VITH NULLSPACEOR CONULL SPACE IN AN IDEAL OF BANACHSPACES Te esaAl a ez 1 .- In oduc ion Le B be he class o all Banach spaces ; he scala ield K is ei he he eal ield o he complex ield . Al1 ope a o s ac ing be ween Banach spaces which appea in his a icle a e supposed o be linea . Fo X,Y e B,£(X,Y) is he space o all ope a o s om X in o Y, he class o all ope a o s om X in o Y wi h dense domain is deno ed by cC D (X,Y), IX deno es he iden i y ope a o on X, J X is he embeddingmap o X in o XII, and X C Y  means ha X is a quo ien space o Y . Fo q T E .C(X,Y), D(T), N(T) and R(T) will deno e he domain, null space and ange o T espec i ely, and we also w i e CON(T) : = Y/R(T), CN(T) : = Y/7), while a(T), S(T) and j(T) will deno e he dimension o N(T), CON(T) and CON(T) espec i ely . We shall conside J6j5(X,Y) : = {T e oC(X,Y) : T is no mally sol able} L(X,Y) : =  (T E £(X,Y) : T is bounded) Le A be an ideal o Banach spaces . Fo in o ma ions and no- a ions abou ope a o ideals and space ideals we e e o [51 . We con- side he ideals, S, R o F, he ideals o all sepa able, e lexi e o ini e dimensional Banach spaces espec i ely . Some no a ions will be used wi hou explana ion because hei meaning is ob ious . In his pape we ob ain a s a e diag am o a linea ope a o wi h dense domain be ween Banach spaces and i s conjuga e ope a o , and we p o e ha his diag am is comple e . 2 . "GENERALIZED" CLASS IFICATION OF (T,T') : STAT E DIAG RAM 2 .1 .  THEOREM .  Le A be an ideal and T E -L D (X,Y) . Then : (i)  U(V) = g(T), a (T) _< B(T') ; in gene al he inequali y is s ic .  I , in addi ion T £ JTs, hen a (T) = d(T') . (ii)  Le A be a comple ely symme ic ideal, hen : CON(T) = Res) °= N(T'), 86 (ii1 ) N(T')(EA i and only i CON(T)EA (ii~ TIE e $ : N(T) £A i and only i CON (T' )C A . (ii3 ) Suppose A su jec i e, i E0N(T')F- A hen N(T)F- A . Fo a bi a y ideals, he p ope ies a e no alid, in gene al . P oo . (i) I is an ob ious consequence o he duali y ela ions, N(T)' = X'/N(T) 0 = (X'/R T'))/(N(T)°/R T')) C C0N(T') To see ha , in gene al, he inequali y a (T) S j(V) is s ic we de ine TeL(1)  by T(a 1 .  a2, . . . . Un . . . .)  _  (0,  a l .  2 1 a2 , . . . . n-1 an  .) . conjuga e ope a o is T'( 0 n ) = (n -1 9n+l )'  ( 0n)F-lm (  U n )e 1 . .I s I is ob ious ha N (T) = (0), 1_ /c, q C CN(T')since R(T')C c, . Clea ly C0N(T') $ R since i m/c,E R henc, has , a subspace isomo phic o l m ; ha con adic s c,e S  and 1 <` S . (ii 1 )  I is su ices o no ice ha N(T') = COÑ (T)' and . ha - Ais comple ely symme ic . (ii 2 )  I Te JdS hen N(T)'  = CON (T') . (ii 3 ) No e ha N(T)' q C CN(T'), A su jec i e and comple ely symme ic . Fo a bi a y ideals, he abo e esul s a e no gua an eed ; o example,  i D : = (X FE B : J XX is complemen ed in X''}  , Tl ,T 2 he null maps on 1, c, espec i ely, hen N(T1) = l m 4 S, CON(T 1 ) = le S N(TZ) = 1(ED, CON(T 2 ) = c,4- D, N(Tl) = 1F-S , CON(T1) = l m 4 S, N(T 2 ) = C .4-D, CQN(T2) = laD . We now in oduce he ollowing classi ica ion o TC .£ (X .Y) . I  :  a (T) <  - . II  : a(T) _- and N(T)E A . III  : a (T) =-and N(T)O A . 1 : B(T)< - . 2 : S(T) = mand -Ñ(T)EA . 3 : B(T) =-and CON(T)$A . By combining hese possibili ies si ua ions . This classi ica ion scheme may conjuga e T' o T . The p ope ies o he (2 .1) heo em on he language o he p e ious classi ica ioncan be w i en as : T'EI~ TE1 T < I T'< 1 TE9$ : Te I 4= :> T'£ 1 A comple ely symme ic : T'EIII41 : :~ TE3 A comple ely A comple ely symme ic and TE," : TEIII4=J T'£3 . We shall p oceed o cons uc a diag am . The shaded in he diag am co espond o s a es ha a e imposible by i ueo (2 .1) heo em . III3 III 2 III 1 II 3 II 2 1 2 symme ic and su jec i e : T E III ==4> TIC 3 1 213  II1  II 2  II3  1111  1112  1113 we ob ainnine di e en now be applied o he squa es ~~II .uS ~S 11 cs ~~ ~QS .Ñ$ ~uS -'~ - ~ . -"' FIEN -" mal cNS' ,NS s1~ 1 , 9, 1 » x@ We analyse i he diag am is comple e, so, we p o e ha a p ocedu e o cons uc s a e exampleso L(X,Y) in he Taylo -Haldbe g classi ica ion, in oduced in 1962 by Goldbe g and Tho p [2], is alid o ou classi ica ion . I El , E2 E B hen he map A  : h E ( El xE2 )'  !>(hl, h 2 )FE i x Ez whe e h 1 (x 1 ) : = h(x l , 0), h2 (x 2 ) : = h(0,  x 2 ),  xi e E i ,  i = 1,2 is an isomo phic,  hus we can iden i y El x E2 e (E l x E2 )' h ough he o mula ($)  (h l , h 2 ) (xl , x2 ): = h(x l , x 2 ): = h l (x 1 ) + h 2 (x2 )E K whe e (x 1 ,  x 2 )E El x E2 . Fo T1 E oC D (X 1' Y1 )' T2 E aC D (X 2 ' Y 2 ) i is possibleiden i y Ti x T2 o (T 1 x T 2 )'  by using he ($) o mula o conside (Ti h 1 , T2 h 2 ) as an elemen o (X 1 x X2 )' .  Also i is clea ha i we de ine he p oduc be ween wo s a eso ou classi ica ion by using he o mula (Aa , Bb ) x (C c ,  Dd ) : =  (max(A,C) max(a,c)' ax(B ' D) max(b,d) ) he s a e o he ope a o T 1 x T2 is he p oduc o he T1 and T 2 s a es . (2 .2) THEOREM . The s a e diag am o (T,T') is comple e . P oo es  : Impossible i A is comple ely symme ic es +05  : Impossible i A is comple ely symme ic and T E -WS es + s  : Impossible i A is comple ely symme ic and su jec i e . ,g$  : Impossible i T E .NS (I 1 , 1 1 ) : Le T be he iden i y ope a o in X . (I 1 , I 2 ) : Le A = R,  (x i ) i6 I  a no malized Hamel basis o 1 2 (N), 88 (e i ) i E I an o hono malbasis o 12 (I) . De ine T :  D(T)C1 2 (I) ---i 1 2 (N) e l -STe i : =,x i whe e D(T) is he linea span o he e,'s . Clea ly D(T) is dense  in  12 (1),  11(1')  =  1 2 (N)  and  N(T)  =  {O ) . Le (enk )kE N C (e i) i El be a sequenceo di e en ec o s and xm : = kil e n /k 2 ED(T), mEN . Then xm--> xn=kElen /k2, k  k Txm -j yn : = k E l  xn /k2 E R(T) i m  hence he e k exis s zn E D(T) such ha Tz n= y n , mo eo e x n -z n ~ 0 since xno D(T), Consequen ly o y E D(T') we ha e ha <xn - zn , T'y > = 0 hus xn - z n E R(V)° . We can choose (e n )  C (ei)iE .I disjoin sequences, k" k EN hus o n E Nwe ob ain (xn - zn) nE Ñ R(T')° ; mo eo e , x n - zn a e linea ly independen , hence dim R T'  ° = m . Clea ly 1 2 (I)/R T'  E R . (1 1' 1 3 )  Le A = R and T be he ope a o in (2 .1) heo em (1 2 ,11 1 )  :  Le A be comple ely symme ic, XEF, YEA - F, T' bé . null map om X in o Y . (1 2 ,111 1 ) :  Le A be non comple ely symme ic, XEF, YEA, Y'4-:A, T he null map om X in o Y (1 3' 11 1 )  Le A be non comple elysymme ic, XEF, Y 4 A, Y'E A, T he null map om X in o Y (1 3 ,111 1 )  :  Le Abe comple ely symme ic, M C X  A, MEA, X/M é .A, T he inclusion om X in o Y (11 1 ,1 2 ) :  In he example (1 2 ,11 1 ) i su ices o eplace T by he conjuga e ope a o . (111 1,1 2 ) :  In he example (1 2 ,111 1 ) i su ices o eplace T by he conjuga e ope a o (III 1 ,1 3 )'  In he example (1 3 ,111 1 ) i su ices o eplace T by he conjuga e ope a o . We can ob ain he emaining allowed s a esby applica ion o he p e ious p ocedu e . S .GOLDBERG . Unbounded linea ope a o s . Mc G aw-Hill,(1966) . S . GOLDBERG, E .0 . THORP . The ange as angespace o compac ope a o s . J . Reine Angew . Ma h ., 211, (1962), 113-115 G .J .O .JAMESON . Topolog y and no med spaces . Chapman and Hall, (1974) . J . LINDENSTRAUSS, L . TZAFRIRI . Classical Banach spaces I, Sp inge -Ve lag, (1977) . A . PIETSCH . Ope a o ideals . No h-Holland, (1980) . [6]  A .E . TAYLOR  C .J .A . HALBERG . Gene a l heo ems abou a bounded linea ope a o and i s conjuga e, J . Reine Angew, Ma h ., 198, (1957), 93-111 . Rebu el 14 de no embne del 1985 Te esa Al a ez Depa amen o de Teo ía de Funciones Facul ad de Ciencias Uni e sidad de San ande San ande , ESPÁÑA REFERENCES