Normal derivations for locally measurable operators

Let a be a normal locally measurable operator affiliated with a semifinite von Neumann algebra M[jls-end-space/]. Let b be a locally measurable operator affiliated with M commuting with a. Then for any locally measurable operator x such that [a,x]+b∈(L1+L∞)(M,τ)[jls-end-space/], we haveb≺≺[a,x]+b, where ≺≺ stands for the Hardy–Littlewood–Pólya submajorization. This extends several results in the existing literature. We also present two applications of the main result. Firstly, we show that the kernels of normal generalized inner derivations on fully symmetric spaces with order continuous norm whose Köthe dual is contained in the ideal of τ-compact operators are orthogonally complemented. Secondly, we establish the regularity of every normal operator acting on a fully symmetric space with order continuous norm, where the norm is not proportional to the Hilbert space norm and the Köthe dual consists of τ-compact operators.