Affordable Access

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

Authors
Publisher
Publicacions de la Secció de Matemàtiques
Publication Date
Source
Legacy

Abstract

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

There are no comments yet on this publication. Be the first to share your thoughts.

Statistics

Seen <100 times
0 Comments
F