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

Pub . Mat . UAB Vol . 30 Nó 1 Maig 1986 STATE DIAGRAM FOR OPERATORS 6VITH NULLSPACEOR CONULL SPACE IN AN IDEAL OF BANACHSPACES TeresaAlvarez 1 .- Introduction Let B be the class of all Banach spaces ; the scalar field K is either the real field or the complex field . Al1 operators acting between Banach spaces which appear in this article are supposed to be linear . For X,Y e B,£(X,Y) is the space of all operators from X into Y, the class of all operators from X into Y with dense domain is denoted by cC D (X,Y), IX denotes the identity operatoron X, J X is the embeddingmap of X into XII, and X C Y  means that X is a quotient space of Y . For q T E .C(X,Y), D(T), N(T) and R(T) will denote the domain, null space and range of T respectively, and we also write CON(T) : = Y/R(T), CN(T) : = Y/7), while a(T), S(T) and j(T) will denote the dimension of N(T), CON(T) and CON(T) respectively . We shall consider J6j5(X,Y) : = {T e oC(X,Y) : T is normally solvable} L(X,Y) : =  (T E £(X,Y) : T is bounded) Let A be an ideal of Banach spaces . For informations and notations about operator ideals and space ideals we refer to [51 . We consider the ideals, S, R or F, the ideals of all separable, reflexive or finite dimensional Banach spaces respectively . Some notations will be used without explanation because their meaning is obvious . In this paper we obtain a state diagram of a linear operator with dense domain between Banach spaces and its conjugate operator, and we prove that this diagram is complete . 2 . "GENERALIZED" CLASS IFICATION OF (T,T') : STAT E DIAG RAM 2 .1 .  THEOREM .  Let A be an ideal and T E -L D (X,Y) . Then : (i)  U(V) = g(T), a (T) _< B(T') ; in general the inequality is strict .  If, in addition T £ JTs, then a (T) = d(T') . (ii)  Let A be a completely symmetric ideal, then : CON(T)t = Res) °= N(T'), 86 (ii1 ) N(T')(EA if and only if CON(T)EA (ii~ TIE ef$ : N(T) £A if and only if CON (T' )C A . (ii3 ) Suppose A surjective, if E0N(T')FA then N(T)FA . For arbitrary ideals, the properties are not valid, in general . Proof . (i) It is an obvious consequence of the duality relations, N(T)' = X'/N(T) 0 = (X'/R T'))/(N(T)°/R T')) C C0N(T') To see that, in general, the inequality a (T) S j(V) is strict we define TeL(1)  by T(a 1 .  a2, . . . . Un . . . .)  _  (0,  a l .  2 1 a2 , . . . . n-1 an  .) . conjugate operator is T'( 0 n ) = (n -1 9n+l )'  ( 0n)F-lm (  U n )e 1 . .Its It is obvious that N (T) = (0), 1_ /c, q C CN(T')since R(T')C c, . Clearly C0N(T')t$ R since if m/c,E R thenc, has , a subspace isomorphic to l m ; that contradicts c,e S  and 1 <` S . (ii 1 )  It is suffices to notice that N(T') = COÑ (T)' and .that - Ais completely symmetric . (ii 2 )  It Te JdS then N(T)'  = CON (T') . (ii 3 ) Note that N(T)' q C CN(T'), A surjective and completely symmetric . For arbitrary ideals, the above results are not guaranteed ; for example,  if D : = (X FE B : J XX is complemented in X''}  , Tl ,T 2 the null maps on 1, c, respectively, then N(T1) = l m 4 S, CON(T 1 ) = le S N(TZ) = 1(ED, CON(T 2 ) = c,4D, N(Tl) = 1F-S , CON(T1) = l m 4 S, N(T 2 ) = C .4-D, CQN(T2) = laD . We now introduce the following classification of TC .£ (X .Y) . I  :  a (T) <  - . II  : a(T) _and N(T)E A . III  : a (T) =-and N(T)OfA . 1 : B(T)< - . 2 : S(T) = mand -Ñ(T)EA . 3 : B(T) =-and CON(T)$A . By combining these possibilities situations . This classification scheme may conjugate T' of T . The properties of the (2 .1) theorem on the language of the previous classificationcan be written as : T'EI~ TE1 T <tI T'<t1 TE9$ : Te I 4= :> T'£ 1 A completely symmetric : T'EIII41 : :~ TE3 A completely A completely symmetric and TE," : TEIII4=J T'£3 . We shall proceed to construct a diagram . The shaded in the diagram correspond to statesthat are imposible by virtueof (2 .1) theorem . III3 III 2 III 1 II 3 II 2 1 2 symmetric and surjective : T E III ==4> TIC 3 1 213  II1  II 2  II3  1111  1112  1113 we obtainnine different now be applied to the squares ~~II .uS ~S 11 cs ~~ ~QS .Ñ$ ~uS -'~ - ~ . -"' FIEN -" mal cNS' ,NS s1~ 1 , 9, 1 » x@ We analyse if the diagram is complete, so, we prove that a procedure to construct state examplesof L(X,Y) in the Taylor-Haldberg classification, introduced in 1962 by Goldberg and Thorp [2], is valid for our classification . If El , E2 E B then the map A  : h E ( El xE2 )'  !>(hl, h 2 )FE i x Ez where h 1 (x 1 ) : = h(x l , 0), h2 (x 2 ) : = h(0,  x 2 ),  xi e E i ,  i = 1,2 is an isomorphic,  thus we can identify El x E2 te (E l x E2 )' through the formula ($)  (h l , h 2 ) (xl , x2 ): = h(x l , x 2 ): = h l (x 1 ) + h 2 (x2 )E K where (x 1 ,  x 2 )E El x E2 . For T1 E oC D (X 1' Y1 )' T2 E aC D (X 2 ' Y 2 ) it is possibleidentify Ti x T2 to (T 1 x T 2 )'  by using the ($) formula to consider (Ti h 1 , T2 h 2 ) as an element of (X 1 x X2 )' .  Also it is clear that if we define the product between two statesof our classification by using the formula (Aa , Bb ) x (C c ,  Dd ) : =  (max(A,C) max(a,c)' ffax(B ' D) max(b,d) ) the state of the operator T 1 x T2 is the product of the T1 and T 2 states . (2 .2) THEOREM . The state diagram for (T,T') is complete . Proof es  : Impossible if A is completely symmetric es +05  : Impossible if A is completely symmetric and T E -WS es + s  : Impossible if A is completely symmetric and surjective . ,g$  : Impossible if T E .NS (I 1 , 1 1 ) : Let T be the identify operator in X . (I 1 , I 2 ) : Let A = R,  (x i ) i6 I  a normalized Hamel basis of 1 2 (N), 88 (e i ) i E I an orthonormalbasis of 12 (I) . Define T :  D(T)C1 2 (I) ---i 1 2 (N) e l -STe i : =,x i where D(T) is the linearspan of the e,'s . Clearly D(T) is dense  in  12 (1),  11(1')  =  1 2 (N)  and  N(T)  =  {O ) . Let (enk )kE N C (e i) i El be a sequenceof different vectors 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) if m  hence there k exists zn E D(T) such that Tz n= y n , moreover x n -z n ~ 0 since xno D(T), Consequently for y E D(T') we have that <xn - zn , T'y > = 0 thus xn - z n E R(V)° . We can choose (e n )  C (ei)iE .I disjoint sequences, k" k EN thus for n E Nwe obtain (xn - zn) nE Ñ R(T')° ; moreover, x n - zn are linearly independent, hence dim R T'  ° = m . Clearly 1 2 (I)/R T'  E R . (1 1' 1 3 )  Let A = R and T be the operator in (2 .1) theorem (1 2 ,11 1 )  :  Let A be completely symmetric, XEF, YEA - F, T'tbé . null map from X into Y . (1 2 ,111 1 ) :  Let A be non completely symmetric, XEF, YEA, Y'4-:A, T the null map from X into Y (1 3' 11 1 )  Let A be non completelysymmetric, XEF, Y 4 A, Y'E A, T the null map from X into Y (1 3 ,111 1 )  :  Let Abe completely symmetric, M C X  A, MEA, X/M é .A, T the inclusion from X into Y (11 1 ,1 2 ) :  In the example (1 2 ,11 1 ) it suffices to replace T by the conjugate operator . (111 1,1 2 ) :  In the example (1 2 ,111 1 ) it suffices to replace T by the conjugate operator (III 1 ,1 3 )'  In the example (1 3 ,111 1 ) it suffices to replace T by the conjugate operator . We can obtain the remaining allowed statesby application of the previous procedure . S .GOLDBERG . Unbounded linear operators . Mc Graw-Hill,(1966) . S . GOLDBERG, E .0 . THORP . The range as rangespace for compact operators . J . Reine Angew . Math ., 211, (1962), 113-115 G .J .O .JAMESON . Topolog y and normed spaces . Chapman and Hall, (1974) . J . LINDENSTRAUSS, L . TZAFRIRI . Classical Banach spaces I, Springer-Verlag, (1977) . A . PIETSCH . Operator ideals . North-Holland, (1980) . [6]  A .E . TAYLOR  C .J .A . HALBERG . Genera l theorems about a bounded linear operator and its conjugate, J . Reine Angew, Math ., 198, (1957), 93-111 . Rebut el 14 de novembne del 1985 Teresa Alvarez Departamento de Teoría de Funciones Facultad de Ciencias Universidad de Santander Santander, ESPÁÑA REFERENCES