An Elementary Product Identity in Polynomial Dynamics

Assuming that this is false, one can choose a subsequence, denoted by u j again, such that ‖u j‖ > j . Let z j := u j/‖u j‖. Then ‖z j‖ = 1, z j is orthogonal to N (I + T ), and z j + T z j = f j/‖u j‖ → 0. As before, it follows that z j → z in H , and passing to the limit in the equation for z j one gets z + T z = 0. Since z is orthogonal to N (I + T ), it follows that z = 0. This is a contradiction since ‖z‖ = lim j→∞ ‖z j‖ = 1. This contradiction proves the desired estimate and the proof is completed. This proof is valid for any compact linear operator T . If T is a finite-rank operator, then the closedness of R(I + T ) follows from a simple observation: finite-dimensional linear spaces are closed.