Invariant subspaces for compact-friendly operators in Sobolev spaces (Q702021)

Let \(S\) and \(T\) be two linear and continuous operators on Sobolev space \(W^{1,p}({\Omega})\). It is said that \(S\) dominates \(T\) if \(| T(u) | \leq S(| u |)\) for every \(u \in W^{1,p}(\Omega)\). A positive operator \(B: W^{1,p}({\Omega}) \rightarrow W^{1,p}({\Omega})\) is called compact-friendly if there exist two nonzero positive operators \(R,K: W^{1,p}({\Omega}) \rightarrow W^{1,p}({\Omega})\) with \(K\) compact in \(W^{1,p}({\Omega})\) such that \(BR=RB\) and \(R\) dominates \(K\). In this paper, the authors investigate compact-friendly operators in Sobolev spaces \(W^{1,p}({\Omega})\). In particular, they show that a compact-friendly operator which is quasinilpotent at a positive vector has a nontrivial closed invariant subspace.