Integer-valued characteristics of solutions of a non-commutative sigma-model

We consider solutions of a noncommutative sigma-model (quantum analogues of harmonic two-spheres in a unitary group) that can be represented as finite-dimensional perturbations of the identity operator in a Hilbert space. Such solutions have three integer-valued characteristics: the normalized energy $e$, the canonical rank $r$, and the minimal uniton number $u$. We prove that always $e\geq r\geq u$ and $2e\geq u(u+1)$ (till now, it was known only that $e\geq r$ and $e\geq u$) and discuss the issue of sufficiency of these inequalities for the existence of a solution with such characteristics.