We prove that if a non-zero weakly compact-friendly operator B on a Banach lattice with topologically full center is locally quasi-nilpotent, then the super right-commutant [B of B has a non-trivial closed invariant ideal. An example of a weakly compact-friendly operator which is not compact-friendly is also provided. | Turk J Math 36 (2012) , 291 – 295. ¨ ITAK ˙ c TUB doi: Invariant subspaces of weakly compact-friendly operators Mert C ¸ a˘glar and Tun¸c Mısırlıo˘glu Abstract We prove that if a non-zero weakly compact-friendly operator B on a Banach lattice with topologically full center is locally quasi-nilpotent, then the super right-commutant [B of B has a non-trivial closed invariant ideal. An example of a weakly compact-friendly operator which is not compact-friendly is also provided. Key Words: Banach lattice, topologically full center, invariant subspace, weakly compact-friendly 1. Introduction Weakly compact-friendly operators have been defined in [3] as a natural extension of compact-friendly operators. Therein, it was shown [3, Theorem ], among others, that a locally quasi-nilpotent weakly compactfriendly operator on a Banach lattice has a non-trivial closed invariant ideal. The purpose of this note is to extend some results in [1] and [4] in the setting of weakly compact-friendly operators on Banach lattices with topologically full center. In doing so, we also provide an example of a weakly compact-friendly operator which is not compact-friendly. Throughout the paper E denotes an infinite-dimensional Banach lattice. As usual, L(E) and L(E)+ stand, respectively, for the algebra of all bounded linear operators and the collection of all positive operators on E . For a positive operator B on a Banach lattice E , the super right-commutant [B of B is defined by [B := {A ∈ L(E)+ | AB − BA ≥ 0}. A subspace V of a Banach space X is called non-trivial if {0} = V = X . If V is a subspace of a Banach lattice and if v ∈ V and |u| ≤ |v| imply u ∈ V , then V is called an ideal. A subspace V of a Banach space X for which T V ⊆ V for a bounded operator T on X is called an invariant subspace for T or a T -invariant subspace. An operator T on E is said to be dominated by a positive operator B on E , denoted by T ≺ B , provided |T x| ≤ B|x| for each x ∈ E . .
