An Application of Kadets-Pełczyński Sets to Narrow Operators

A known analogue of the Pitt compactness theorem for function spaces asserts that if 1 ≤ p < 2 and p < r < ∞, then every operator T : Lp → Lr is narrow. Using a technique developed by M.I. Kadets and A. Pełczyński, we prove a similar result. More precisely, if 1 ≤ p ≤ 2 and F is a Köthe {Banach space on [0; 1] with an absolutely continuous norm containing no isomorph of Lp such that F is subset of Lp, then every regular operator T : Lp → F is narrow.Известный аналог теоремы Питта о компактности для функциональных пространств утверждает, что если 1 ≤ p < 2 и p < r < ∞, то каждый оператор Lp → Lr узкий. Используя технику, разработанную М.И. Кадецем и А. Пелчинским, мы доказываем похожий результат. Именно, если 1 ≤ p ≤ 2 и F - банахово пространство Кете на [0; 1] с абсолютно непрерывной нормой, не содержащее подпространств, изоморфных Lp, причем F является подмножеством Lp, то каждый регулярный оператор T : Lp → F узкий

Замiтка про оператори з функцiональних просторiв Кете у простiр c0(Γ)c_0(\Gamma)

It is well known that every operator from $E = L_p$, 1 \leq p < \infty to $c_0$ is narrow. We show that this result can be extended to a more general class of Köthe function spaces $E$.Метою замітки є узагальнення відомого результату про вузькість будь-якого оператора з простору $E = L_p$ в $c_0$ при 1 \leq p < \infty на випадок загальнішого класу просторів Кете $E$

A note on operators from K"{o}the function spaces to c0(Gamma)c_0(Gamma)

It is well known that every operator from $E = L_p$, $1 leq p <infty$ to $c_0$ is narrow. We show that this result can beextended to a more general class of K"{o}the function spaces$E$

Points of narrowness and uniformly narrow operators

It is known that the sum of every two narrow operators on $L_1$ is narrow, however the same is false for $L_p$ with $1 0$ there exists a decomposition $e = e' + e''$ to disjoint elements such that $\|S(e') - S(e'')\| < \varepsilon$ and $\|T(e') - T(e'')\| < \varepsilon$. The standard tool in the literature to prove the narrowness of the sum of two narrow operators $S+T$ is to show that the pair $S,T$ is uniformly narrow. We study the question of whether every pair of narrow operators with narrow sum is uniformly narrow. Having no counterexample, we prove several theorems showing that the answer is affirmative for some partial cases