Properties of modulus of monotonicity and Opial property in direct sums

Abstract

We give an example of a Banach lattice with a non-convex modulus of monotonicity, which disproves a claim made in the literature. Results on preservation of the non-strict Opial property and Opial property under passing to general direct sums of Banach spaces are established.

Opial properties in interpolation spaces

“…where 𝜖 ∈ [0, 1]. The function 𝛿 𝑚,𝐸 is increasing and continuous in [0,1), but not necessarily convex (see [19]). A Banach lattice 𝐸 is uniformly monotone if 𝛿 𝑚,𝐸 (𝜖) > 0 for every 𝜖 > 0.…”