The paper gives i t characterization of right-hand quotients d = Al o -4;' of surjective operat,or ideals A, iiiid 112 which refers to so cnlled "generitting systems of sets". Siinultmeously the corresponding "inversion formulas" il, = ( A o A,)S and A, = 11-1 o Ai iwe investigated.
0. Basic concepts and notations
The present paper deals with a special method for geneiating o p n t o r ideals o ? ~ the class of all UANACH spaces. The general riotion of an operator ideal can he looked np in [lo] and in various research papers.
The teriii "operator" refer? to a eontinuous linear map froiii a BANACH space I.J into a BANACH space F . The class of all these iiiaps is denoted by L(E, F ) . The notation
A ( E , F ) concerns the class of operators T E L(E, F ) belorigiiig to a given operator ideal A .
The rig?![-hand quotient A, o A;, of two operator ideals A, rmcl A, is defined to lw the class of operators T such that T S belongs to A, for all X of type A,. Similarly the
left-htrnd yuotzejit
o A, is meant to he the class of operators T such that AST 1s of type 9, for all S of type A,. Finally the ptoduct A, o A, of two opercctor ideuls A, and A, qtaiids for the class of operators T which can be represented by a product T = BS, $liere R is of type A, and S of type A,. Quotients and products of operator ideal? again turn out to he operator ideals (cf. [lo], [HI). The notion of r( surjectave opetator ideutl, which is basic for this paper, has hcc.11 introduced in [14] and [15]. Roughly speaking, surjectivity of an operator ideal ,1 tiieans that the operators 7' E A ( K , F ) are characterized by the nature of the image 7'(UE) they iiiake from the unit hall UE of the BANACH space E'. Indeed, denoting the closed unit balls of any BANACH spaces E and E , hy U, and UEo, respectively, e ('itii descrihe a u r p t iziity of u 71 operutor deal A as follows : An cyiiivalent cliaracterizatioii of surjectivity says that Given HI^ arhit.mry opcrator idcal A the .wrjective hull As is meant to he the crlnss of operator# 7' which ace siibject to an inclusion 7'( u&) c AS( ( I / & ) , 8 being an operator of tyim A, say 8 E A(&,,, B') a n d ' 1' c / A ( & , J'). Another ehmicterizcriion of the 'turjecfive hull AS refers to (g'), nauiely: 2' E L(E, F ) be1on.p to AS(E, F ) if { E n d on49 if there i s (4 nurjmtion 0 : P . + F: ,WCJI that 'I'Q f A(,!& 3'). The aurjcctive hull As ir, the stnallest surjective opcrator ideal containing A. Ei,r couples (9, P), where E = Z l,(r) is the BANACH space of a11 stimiuable nuriibec faniilicn (Ay}yc,on t.he index set I' and where E' ist an arbitrary BAXACH Rpace, we have coincidence (cf. [14'1, [lS]).