Year: 2024
Type: article
Abstract: Let ( Ω i , ∑ i , μ i ) , i = 1, 2 be two measure spaces, 1< p < ∞ , X = L p ( Ω 1 , ∑ 1 , μ 1 ) , Y = L p ( Ω 2 , ∑ 2 , μ 2 ) , and let f : X → Y be a standard ε-norm-additive mapping, i.e., f ( 0 ) = 0 and | ∥ f ( x ) + f ( y ) ∥ − ∥ x + y ∥ | ≤ ε , ∀ x , y ∈ X . In this paper, we show that if f : X → Y is a non-surjective standard ε-norm-additive mapping, then there exists a bounded linear surjective operator T : Y → X with ∥ T ∥ = 1 such that ‖ T f ( x ) − x ‖ ≤ 3 2 ε , ∀ x ∈ X . Moreover, if f is almost surjective, then there exists a linear surjective isometry U : X → Y such that ‖ f ( x ) − U ( x ) ‖ ≤ 3 2 ε , ∀ x ∈ X . The estimates “ 3 2 ε ” in above results are optimal.
Source: Scientia Sinica Mathematica
Author: Longfa Sun
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W4404495320