Domination problems for strictly singular operators and other related classes (original) (raw)

On positive strictly singular operators and domination

Positivity, 2003

We study the domination problem by positive strictly singular % operators between Banach lattices. Precisely we show that if E and %F are two Banach lattices such that the norms on E' and F are %order continuous and E satisfies the subsequence splitting property, %and %0≤S≤ T : E → F are two positive operators, then T strictly %singular implies S strictly singular. The special case of %endomorphisms is also considered. Applications to the class of %strictly co-singular (or Pelczynski) operators are given too.

DOMINATION BY POSITIVE DISJOINTLY STRICTLY SINGULAR OPERATORS

2000

We prove that each positive operator from a Banach lattice E to a Banach lattice F with a disjointly strictly singular majorant is itself disjointly strictly singular provided the norm on F is order continuous. We prove as well that if S : E → E is dominated by a disjointly strictly singular operator, then S 2 is disjointly strictly singular.

POWERS OF OPERATORS DOMINATED BY STRICTLY SINGULAR OPERATORS

Quarterly Journal of Mathematics, 2007

It is proved that every positive operator R on a Banach lattice E dominated by a strictly singular operator T : E → E satisfies that the R 4 is strictly singular. Moreover, if E is order continuous then the R 2 is already strictly singular.

Domination by Positive Narrow Operators

Positivity, 2003

We prove that each positive operator from a Köthe function-space E(μ) to a Banach lattice F with a narrow majorant is itself narrow provided the norm on F is order continuous. We also prove that every l 2-strictly singular regular operator from L p[0,1], 1≤p F is narrow, provided F has an order continuous norm.

Domination by positive Banach–Saks operators

Studia Mathematica, 2006

Given a positive Banach-Saks operator T between two Banach lattices E and F , we give sufficient conditions on E and F in order to ensure that every positive operator dominated by T is Banach-Saks. A counterexample is also given when these conditions are dropped. Moreover, we deduce a characterization of the Banach-Saks property in Banach lattices in terms of disjointness.

SOME PROPERTIES OF STRICTLY SINGULAR OPERATORS ON BANACH LATTICES

maia.ub.es

Several results obtained during the author's Ph.D. Thesis are presented. In particular, domination results (in Dodds-Fremlin sense) for the ideal of strictly singular operators will be given. Moreover, the connections between strictly singular and the classes of AM-compact, l2-singular and disjointly strictly singular are studied. As an application we obtain existence of invariant subspaces for positive strictly singular operators. On a di erent direction, results on compact powers of strictly singular operators are also presented extending a theorem of V. Milman. Finally, we study when a c0-singular or l1-singular operator can be extended to an operator between vector valued lattices preserving its singularity properties.

Positivity Schur operators and domination problem

In this paper we are concerned with developing generalizing concepts of Dunford–Pettis operators analogous to the generalization of compact operators by strictly singular operators. Also, we give some new results concerning the domination problem in the setting of positive operators between Banach lattices.

Factorization and domination of positive Banach-Saks operators

Studia Math, 2008

It is proved that every positive Banach-Saks operator T : E → F between Banach lattices E and F factors through a Banach lattice with the Banach-Saks property, provided that F has order continuous norm. By means of an example we show that this order continuity condition cannot be removed. In addition, some domination results, in the Dodds-Fremlin sense, are obtained for the class of Banach-Saks operators.

Domination properties in ordered Banach algebras

Studia Mathematica, 2002

We recall from [9] the definition and properties of an algebra cone C of a real or complex Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A. The Banach algebra A is then called an ordered Banach algebra. An important property that the algebra cone C may have is that of normality. If C is normal, then the order structure and the topology of A are reconciled in a certain way. Ordered Banach algebras have interesting spectral properties. If A is an ordered Banach algebra with a normal algebra cone C, then an important problem is that of providing conditions under which certain spectral properties of a positive element b will be inherited by positive elements dominated by b. We are particularly interested in the property of b being an element of the radical of A. Some interesting answers can be obtained by the use of subharmonic analysis and Cartan's theorem.