Given a constant q ∈ (1, ∞), we study the following property of a normed sequence space E: If {x ... more Given a constant q ∈ (1, ∞), we study the following property of a normed sequence space E: If {x n } n∈N is an element of E and if {y n } n∈N is an element of ℓ q such that ∞ n=1 |x n | q = ∞ n=1 |y n | q and if the nonincreasing rearrangements of these two sequences satisfy N n=1 |x * n | q ≤ N n=1 |y * n | q for all N ∈ N, then {y n } n∈N ∈ E and {y n } n∈N E ≤ C {x n } n∈N E for some constant C which depends only on E. We show that this property is very close to characterizing the normed interpolation spaces between ℓ 1 and ℓ q. More specificially, we first show that every space which is a normed interpolation space with respect to the couple (ℓ p , ℓ q) for some p ∈ [1, q] has the above mentioned property. Then we show, conversely, that if E has the above mentioned property, and also has the Fatou property, and is contained in ℓ q , then it is a normed interpolation space with respect to the couple ℓ 1 , ℓ q. These results are our response to a conjecture of Galina Levitina, Fedor Sukochev and Dmitriy Zanin in [11].
Given a constant q ∈ (1, ∞), we study the following property of a normed sequence space E: If {x ... more Given a constant q ∈ (1, ∞), we study the following property of a normed sequence space E: If {x n } n∈N is an element of E and if {y n } n∈N is an element of ℓ q such that ∞ n=1 |x n | q = ∞ n=1 |y n | q and if the nonincreasing rearrangements of these two sequences satisfy N n=1 |x * n | q ≤ N n=1 |y * n | q for all N ∈ N, then {y n } n∈N ∈ E and {y n } n∈N E ≤ C {x n } n∈N E for some constant C which depends only on E. We show that this property is very close to characterizing the normed interpolation spaces between ℓ 1 and ℓ q. More specificially, we first show that every space which is a normed interpolation space with respect to the couple (ℓ p , ℓ q) for some p ∈ [1, q] has the above mentioned property. Then we show, conversely, that if E has the above mentioned property, and also has the Fatou property, and is contained in ℓ q , then it is a normed interpolation space with respect to the couple ℓ 1 , ℓ q. These results are our response to a conjecture of Galina Levitina, Fedor Sukochev and Dmitriy Zanin in [11].
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