Generating Systems of Sets and Quotients of Surjeetive Operator Ideals

Abstract

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" Basic concepts and notationsThe 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

$${\mathcal A}$$ A -compact mappings

“…Motivated by this work, Ryan [23] carried out a similar study of weakly compact holomorphic mappings, obtaining similar result to those in [6]. In 2000, González and Gutiérrez [17], using the theory of generating system of sets of Stephani [25], extend some of the results of [6] and [23] to a wide class of holomorphic mappings.…”

“…Also, we say that A is surjective if A = A sur isometrically. For a Banach operator ideal A, a set M ⊂ X is A-bounded if there exist a Banach space Z and a linear operator T ∈ A(Z; Y ) such that M ⊂ T (B Z ), see [25]. In the case of A = W, the W-bounded sets coincide with the weakly compact sets and if A = K, we obtain the compact sets.…”

Ideal topologies and corresponding approximation properties

“…Proposition 2.4 can be used to provide many further useful examples of ideal topologies. Given an operator ideal I and a Banach space E, according to [63,28,37] we define…”

Topological and bornological characterisations of ideals in von Neumann algebras: I

“…(We will not be any more precise than this here, simply referring the reader to [4] w167 4.7.) We will show that ideals in a yon Neumann algebra satisfy properties analogous to certain charaeterisations of injective and surjective operator ideals studied by Stephani ([6], [7] and [8]). As a consequence we show that some of the known properties of injective and surjective operator ideals have analogues in the setting of ideals in yon Neumann algebras.…”

“…Dual to the notion of 'generating topology' is Stephani's 'generating systems of sets'. In [7] it is shown that if an operator ideal is surjective then it can be characterised in terms of these generating systems of sets. This result can be compared to the bornological characterisation mentioned above.…”