Abstract. We prove for a large class of operators J, including block Jacobi matrices, that if .J / n OE˛;ˇ is a finite set, each point of which is an eigenvalue of finite multiplicity, then a finite coefficient stripped, J N , has .J N / OE˛;ˇ. We use an abstract Dirichlet decoupling.Mathematics Subject Classification (2010). 47B26, 34L15, 81Q10.