Finite elements in some spaces of operators

The main result of this thesis asserts that all lattice norm 𝗌 𝗍𝗁𝖺𝗍 make a vector lattice into a Banach lattice 𝖺𝗋𝖾 equivalent. Also, for any D𝖾𝖽𝖾𝗄𝗂𝗇𝖽 complete Banach latt 𝕏, then φ1(ℒr(𝕏)) = ℒr(𝕏) if and only if 𝑑𝑖𝑚𝕏 < ∞. Let 𝕏 and 𝕐 be two Banach lattices and 𝕏 be an 𝐴𝐿 - space, 𝕐 D𝖾𝖽𝖾𝗄𝗂𝗇𝖽 complete with an 𝗈𝗋𝖽𝖾𝗋 unit. If 𝜑1(ℒ𝑟(𝕏, 𝕐)) = ℒ𝑟(𝕏, 𝕐), then 𝕏 is lattice isomorphic 𝗍𝗈 an 𝐴𝐿-space and 𝕐 is lattice isomorphic to an 𝐴𝑀-space. Under the same assertions ℒ𝑟(𝕏, 𝕐) has a rank one operator as an 𝚘rd𝚎r unit. If for any 𝒽 ∈ 𝜑1(𝕏′) and for any 𝑦 ∈ 𝕐, the rank one operator𝒽 ⊗ 𝑦 is a 𝖿𝗂𝗇𝗂𝗍𝖾 element in 𝒦(𝕏, 𝕐). 2023, 44 pages Keywords: ordered vector space, Archimedean ordered vector space, Selfmajorizing element, Lattic ısomorphism, F𝗂𝗇𝗂𝗍𝖾 element