Structure of the canonical uniton factorization of a solution of a noncommutative unitary sigma model

It is known that each solution $\Phi$ with a nonzero finite energy can be represented up to a multiplicative constant as a composition of finitely many reflections of the special form $\Phi = e^{i\theta}(I-2P_1) \dots (I-2P_n)$. This representation is called the canonical uniton factorization. Orthogonal projections $P_1, \dots, P_n$, called unitons, have finite-dimensional images $\alpha_1, \dots, \alpha_n$. We show that for $1\le j\le n$, the subspaces $\alpha_1+\dots+\alpha_j$ are invariant under the annihilation operator, and the annihilation operator eigenvalues coincide on these subspaces.