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13 pages, 274 KiB

Open AccessArticle

Upper Local Uniform Monotonicity in F-Normed Musielak–Orlicz Spaces

by Yanli Liu, Yangyang Xue and Yunan Cui

Axioms 2023, 12(6), 539; https://doi.org/10.3390/axioms12060539 - 30 May 2023

Viewed by 1158

In this paper, the necessary and sufficient conditions for the upper strict monotonicity point and the upper local uniform monotonicity point are given in the case of Musielak–Orlicz spaces equipped with the Mazur–Orlicz F-norm. Moreover, strict monotonicity and upper local uniform monotonicity are easily deduced in the case of Musielak–Orlicz spaces endowed with the Mazur–Orlicz F-norm, and the work by Kaczmarek presented in the references is encompassed by the corollaries presented in this paper. Full article

(This article belongs to the Special Issue Modern Functional Analysis and Related Applications)

12 pages, 293 KiB

Open AccessArticle

On Bilinear Narrow Operators

by Marat Pliev, Nonna Dzhusoeva and Ruslan Kulaev

Mathematics 2021, 9(22), 2892; https://doi.org/10.3390/math9222892 - 13 Nov 2021

Cited by 2 | Viewed by 1696

Abstract

In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator

T : E \times F \to W

defined on the Cartesian product of vector lattices E and F and taking [...] Read more.

In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator

T : E \times F \to W

defined on the Cartesian product of vector lattices E and F and taking values in a vector lattice W is narrow if the partial operators

T_{x}

and

T_{y}

are narrow for all

x \in E, y \in F

. We prove that, for order-continuous Köthe–Banach spaces E and F and a Banach space X, the classes of narrow and weakly function narrow bilinear operators from

E \times F

to X are coincident. Then, we prove that every order-to-norm continuous C-compact bilinear regular operator T is narrow. Finally, we show that a regular bilinear operator T from the Cartesian product

E \times F

of vector lattices E and F with the principal projection property to an order continuous Banach lattice G is narrow if and only if

| T |

is. Full article

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