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Domination by Positive Strictly Singular Operators
Author(s) -
Flores Julio,
Hernández Francisco L.
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003447
Subject(s) - mathematics , compact operator , compact space , finite rank operator , strictly singular operator , pure mathematics , operator (biology) , norm (philosophy) , lattice (music) , order (exchange) , discrete mathematics , banach space , endomorphism , combinatorics , quasinormal operator , extension (predicate logic) , physics , computer science , repressor , law , chemistry , acoustics , biochemistry , political science , transcription factor , programming language , finance , economics , gene
The problem of domination for positive compact operators between Banach lattices was solved by Dodds and Fremlin in [ 6 ]: given a Banach lattice E with order continuous dual norm, an order continuous Banach lattice F and two positive operators 0 ⩽ S ⩽ T : E → F , the operator S is compact if T is. A similar problem has been considered in the class of weakly compact operators by Abramovich [ 1 ] and in a general form by Wickstead [ 26 ]. Precisely, Wickstead's result shows that the operator S is weakly compact if T is whenever one of the following two conditions holds: either E ′ is order continuous or F is order continuous. When it comes to Dunford–Pettis operators, Kalton and Saab [ 15 ] have proved that the operator S is Dunford–Pettis if T is, provided that the Banach lattice F is order continuous. On the other hand, Aliprantis and Burkinshaw settled the problem of domination for compact [ 2 ] and weakly compact [ 3 ] endomorphisms (that is, the case when E = F ). For example, they proved that if either the norm on E or the norm on E ′ is order continuous, then the compactness of T is inherited by the power operator S 2 . Also, they showed that, for E an arbitrary Banach lattice, T being compact always implies that S 3 is compact. More recently, Wickstead studied converses for the Dodds–Fremlin and Kalton–Saab theorems in [ 27 ] and for the Aliprantis–Burkinshaw theorems in [ 28 ].