Unbounded order continuous operators on Riesz spaces

Abstract

In this paper we introduce two new classes of operators that we call strongly order continuous and strongly σ-order continuous operators. An operator T : E → F between two Riesz spaces is said to be strongly order continuous (resp. strongly σ-orderWe give some conditions under which order continuity will be equivalent to strongly order continuity of operators on Riesz spaces. We show that the collection of all so-continuous linear functionals on a Riesz space E is a band of E ∼ .

Unbounded order-norm continuous and unbounded norm continuous operator

“…The notion of unbounded order convergence (uo-convergence, for short) was firstly introduced by Nakano in [17]. After that, Bahramnezhad and Haghnejad Azar proposed the definition of unbounded order continuous operators in [4]. It is clear that for order bounded nets, uo-convergence is equivalent to oconvergence.…”

Continuous operators for unbounded convergence in Banach lattices

“…Recall that an operator T from a Banach space X to a Banach space Y is DunfordPettis if it maps weakly null sequences of X to norm null sequences of Y and is weak DunfordPettis if f n (T (x n )) → 0 for any w-null sequence (x n ) in X and any w-null sequence (f n ) in Y . An operator T between two Riesz space is called unbounded order continuous operator whenever x α uo − → 0 implies T x α uo − → 0 [9]. An operator T from a Banach lattice E into a Banach space X is said to…”

“…The compact operators based on unbounded convergence has been investigated in [7,10]. Recently, unbounded order continuous operator and uaw-Dunford-Pettis operator are also studied in [9,10,11]. Now, we consider the continuous operators based on unbounded topology convergence.…”

Order Continuous Operators

“…Many researchers studied this topic, and some of these previous studies have proven the theory of extending the positive order continuous operators (Veksler, 1960). Another study defined new classes of operators, which are socalled unbounded order continuous and further boundedly unbounded order continuous operators and gave extra settings under which uo-continuity is equivalent to order continuity of some operators on Riesz spaces (Bahramnezhad & Azar, 2018). The report investigated the relationships located between order to topology continuous operators and different types of operators for example b-weakly compact, order weakly compact and order continuous operators, and studied adjoint of order to norm continuous operators (Jalili et al, 2021).…”