Harmonic maps and shift-invariant subspaces

With the help of operator-theoretic methods, we derive new and powerful criteria for finiteness uniton number a harmonic map from Riemann surface to unitary group $${{\,\mathrm{U}\,}}(n)$$ . These use Grassmannian model where maps are represented by families shift-invariant subspaces $$L^2(S^1,{{\mathbb {C}}}^n)$$ ; give description that model.