Given an operator T on Hilbert space H, and an element y in H, we construct a compact, positive selfadjoint operator \(W_y\) on H. In [1], \(W_y\) is used to produce non-cyclic vectors—invariant subspaces—for T. In this note—observing that \(\alpha T\) has the same invariant subspaces as T—we study the behavior of \(W_y\) , when \(W_y\) is constructed from the operator \(\alpha T\) as \(\alpha \to 0\) .