On Some Linear Operators Preserving Disjoint Support Property
Sahand Communications in Mathematical Analysis
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Abstract
The aim of this work is to characterize all bounded linear operators T : ℓ p (I) → ℓ p (I) which preserve disjoint support property. We show that the constant coefficients of all isometries on ℓ p (I) are in the class of such operators, where 2 ̸ = p ∈ [1, ∞) and I is a non-empty set. We extend preserving disjoint support property to linear operators on c0(I). At the end, we obtain some equivalent properties of isometries on Banach spaces.
Key takeaways
- In this work, we characterize all bounded linear operators on ℓ p (I) and c 0 (I) which preserve disjoint support property and obtain some properties of them, where I is a non-empty set.
- Some examples of preserving disjoint support linear operators on ℓ p are presented.
- Throughout this section, I is a non-empty set, p ∈ [1, +∞), and ℓ p (I) is the Banach space of all functions f : I → R with the finite norm defined by
- We call T preserving disjoint support property, if for all f, g ∈ ℓ p (I) the equality f g = 0 implies T f T g = 0.
- We proceed with the study of the bounded linear operators T : c 0 (I) → c 0 (I) which are in P ds (c 0 (I)), and we prove that all rows of the matrix form of such operators belong to ℓ 1 (I).
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