Weakly compact-friendly operators

We introduce weak compact-friendliness as an extension of compact-friendliness, and and prove that if a non-zero weakly compact-friendly operator $B\colon E\to E$ on a Banach lattice is quasi-nilpotent at some non-zero positive vector, then $B$ has a non-trivial closed invariant ideal. Relevant facts related to compact-friendliness are also discussed.