Hausdorff sets of matrices and an estimate for the angle between invariant subspaces (Q1921797)

Let \(A: \mathbb{C}^n\to\mathbb{C}^n\) be a linear operator and assume \(U_1\) and \(U_2\) are two invariant subspaces of \(A\). We define the following sets \[ H=\Biggl\{{(Au,u)\over(u,u)}: u\in\mathbb{C}^n\Biggr\},\quad H_j=\Biggl\{{(Au,u)\over(u,u)}: u\in U_j\Biggr\},\quad j=1,2. \] For \(j=1,2\) let \(d_j\) be the diameter of \(H_j\), \(l\) be the distance of \(H_1\) and \(H_2\), \[ L=\min_{z_j\in H_j,z\in\partial H}(|z-z_1|+|z-z_2|),\quad\rho= \max_{z_1\in H_1}\Biggl(\min_{z\in\partial H}(|z-z_1|)\Biggr). \] We define the angle \(\varphi= \varphi(U_1,U_2)\) between the two subspaces as follows \[ \varphi(U_1,U_2)= \min_{u\in H_1,v\in H_2}\text{arccos }{|(u,v)|\over|u||v|}. \] In the present article it is shown that \[ \sin\varphi(U_1,U_2)\geq {1\over L+d_1+d_2},\quad\sin\varphi(U_1, U_2)\geq{1\over 1+2\rho/\ell}. \] The authors have been motivated by the work of \textit{A. S. Markus} [Dokl. Akad. Nauk SSSR 132, 524-527 (1960; Zbl 0095.31203); Engl. transl.: Soviet Math., Dokl. 1, 599-602 (1960)] concerning the estimates for the angles between root vectors of a dissipative operator.