Multiplication operators in BV spaces

Reinwand, Simon; Kasprzak, Piotr

doi:10.1007/s10231-022-01260-4

Cited by 2 publications

(3 citation statements)

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“…Example 1 (cf. [5], Example 12). Consider the space RBV p ([0, 1]) of functions with Riesz finite p-variation (for some p ≥ 1), i.e., with finite seminorms:…”

Section: Norms On Subalgebrasmentioning

confidence: 99%

“…However, the previous form of the lemma required a condition that was quite difficult to evaluate: hence, the restriction to the space of functions with bounded variation-although considered in a very different sense (cf. [4,5]). For example, in a recent paper [6], this approach is used to study sequences of generalized bounded variation.…”

Section: Introductionmentioning

confidence: 99%

“…The question was posed in the work of [1], who studied many problems concerning the space of functions with generalized bounded variation; ([4,13]) concerns the construction of a norm to ensure continuity of multiplication in this norm and its application: for example, in quadratic operator equations. It is also important in the study of multiplier spaces (for example, [5]). Given a subspace (X, ∥ • ∥ X ) of the space of bounded functions, we have a question: how to find a norm ∥ • ∥ that is equivalent to ∥ • ∥ X and for which (X, ∥ • ∥) is a normalized normed algebra, i.e.,:…”

Section: Introductionmentioning

confidence: 99%

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On Normed Algebras and the Generalized Maligranda–Orlicz Lemma

Cichoń,

Cichoń

2023

Symmetry

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In this paper, we discuss some extensions of the Maligranda–Orlicz lemma. It deals with the problem of constructing a norm in a subspace of the space of bounded functions, for which it becomes a normed algebra so that the norm introduced is equivalent to the initial norm of the subspace. This is done by satisfying some inequality between these norms. We show in this paper how this inequality is relevant to the study of operator equations in Banach algebras. In fact, we study how to equip a subspace of the space of bounded functions with a norm equivalent to a given one so that it is a normed algebra. We give a general condition for the construction of such norms, which allows us to easily check whether a space with a given norm is an algebra with a pointwise product and the consequences of such a choice for measures of noncompactness in such spaces. We also study quasi-normed spaces. We introduce a general property of measures of noncompactness that allows the study of quadratic operator equations, prove a fixed-point theorem suitable for such problems, and complete the whole with examples and applications.

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“…Example 1 (cf. [5], Example 12). Consider the space RBV p ([0, 1]) of functions with Riesz finite p-variation (for some p ≥ 1), i.e., with finite seminorms:…”

Section: Norms On Subalgebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%