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

Publicacions de la Secció de Matemàtiques
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Pub . Mat . UAB Vol . 30 Nó 1 Maig 1986 STATE DIAGRAM FOR OPERATORS 6VITH NULL SPACE OR CONULL SPACE IN AN IDEAL OF BANACH SPACES Teresa Alvarez 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 cCD (X,Y), IX denotes the identity operator on X, JX is the embedding map of X into XII, and X C Y means that X is a quotient space of Y . Forq 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 no- tations about operator ideals and space ideals we refer to [51 . We con- sider 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') : STATE DIAGRAM 2 .1 . THEOREM . Let A be an ideal and TE -LD(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 s

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