Canonical angles of unitary spaces and perturbations of direct complements (Q1301314)

The theorem on the nontrivial comlementarity of subspaces of \(\mathbb{C}^n\) by \textit{E. Berkson} [Pac. J. Math. 13, 7-22 (1963; Zbl 0118.10402)] and \textit{J. M. Schumacher} [Syst. Control Lett. 18, No. 1, 1-8 (1992; Zbl 0743.93080)] determining conditions for a subspace \({\mathcal M}\) to be complementary to \({\mathcal W}\) is extended. Bounds \(r_k\), \(1\leq k\leq r\) such that \(\theta({\mathcal L}, {\mathcal M})< r_k\) implies \(\dim ({\mathcal M}\cap {\mathcal W})< k\), are found. The bounds \(r_k\) are expressed in terms of angles between \({\mathcal L}\) and \({\mathcal W}\).