Unbounded order-to-order continuous operators and order-to-unbounded order continuous operators on Riesz spaces Unbounded Order-to-Order Continuous Operators on Riesz Spaces

Let E and F be two Riesz spaces. An operator T : E→ F between two Riesz spaces is said to be unbounded order-to-order continuous whenever x∝→ ۰ in E implies Tx∝ → ۰ in F for each net (x∝)⊆ E. This paper aims to investigate several properties of a novel class of operators and their connections to established operator classifications. Furthermore, we introduce a new class of operators, which we refer to as order-to-unbounded order continuous operators. An operator T : E→ F rightarrow F between two Riesz spaces is said to beorder-to-unbounded order continuous (for short, ouo-continuous), if x∝→ ۰ in E implies Tx∝ → ۰ in F for each net (x∝)⊆ E.In this manuscript, we investigate the lattice properties of a certain class of objects and demonstrate that, under certain conditions, order continuity is equivalent to unbounded order-to-order continuity of operators on Riesz spaces. Additionally, we establish that the set of all unbounded order-to-order continuous linear functionals on a Riesz space E forms a band of E∼.

Keywords:

Riesz space , order convergence , unbounded order convergence