On the numerical index of real Lₚ(μ)-spaces

الأصل · EN

We give a lower bound for the numerical index of the real space Lₚ(μ) showing, in particular, that it is non-zero for p≠ 2. In other words, it is shown that for every bounded linear operator T on the real space Lₚ(μ), one has |∫ |x|ᵖ⁻¹(x) T x dμ|: x∈ Lₚ(μ), x=1 ≥ Mₚ/12T where Mₚ=ₜ∈[₀,₁]|tᵖ⁻¹-t|1+tᵖ>0 for every p≠ 2. It is also shown that for every bounded linear operator T on the real space Lₚ(μ), one has ∫ |x|ᵖ⁻¹|Tx| dμ: x∈ Lₚ(μ), x=1 ≥ 1/2T.