order-norm continuous operators and order weakly compact operators

2022/10/21 by Sajjad Ghanizadeh Zare, Kazem Haghnejad Azar, Zare, Sajjad Ghanizadeh +5
Computer Science · Mathematics · #46B40 #46B42 #47B65 #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2210.13975

openalex publication_date 2022/10/21 · openalex created_date 2022/11/01 · openalex updated_date 2026/07/28

Abstract

Let E be a sublattice of a vector lattice F. A continuous operator T from the vector lattice E into a normed vector space X is said to be order-norm continuous whenever x_α\xrightarrowFo0 implies Tx_α\xrightarrow\Vert.\Vert0 for each (x_α)_α⊆ E. Our mean from the convergence x_α\stackrelFo \longrightarrow x is that there exists another net (y_α) in F with the same index set satisfying y_α\downarrow 0 in F and \vert x_α- x \vert ≤ y_α for all indexes α. In this paper, we will study some properties of this new class of operators and its relationships with some known classifications of operators. We also define the new class of operators that named order weakly compact operators. A continuous operator T: E → X is said to be order weakly compact, if T(A) in X is a relatively weakly compact set for each Fo-bounded A⊆ E. In this manuscript, we study some properties of this class of operators and its relationships with order-norm continuous operators.

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