There is no strictly singular centralizer on 𝐿_{𝑝}

Abstract

We prove that if Φ \Phi is a centralizer on L p L_p , where 0 > p > ∞ 0>p>\infty , then there is a copy of ℓ 2 \ell _2 inside L p L_p where Φ \Phi is bounded. If Φ \Phi is symmetric, then it is also bounded on a copy of ℓ q \ell _q , provided 0 > p > q > 2 0>p>q>2 . This shows that for a wide class of exact sequences 0 → L p → Z → L p → 0 0\to L_p\to Z\to L_p\to 0 the quotient map is not strictly singular, which generalizes a recent result of Jesús Suárez.