Systems of subspaces in Hilbert space that obey certain conditions, on their pairwise angles

Abstract:

The paper is devoted to systems of subspaces $H_{1},\dots ,H_{n}$ of a complex Hilbert space $H$ that satisfy the following conditions: for every index $i>1$, the angle $\theta _{1,i}\in (0,\pi /2)$ between $H_{1}$ and $H_{i}$ is fixed; the projections onto $H_{2k}$ and $H_{2k+1}$ commute for $1\leq k\leq m$ ($m$ is a fixed nonnegative number satisfying $m\leq (n-1)/2$); all other pairs $H_{i}$, $H_{j}$ are orthogonal. The main tool in the study is a construction of a system of subspaces in a Hilbert space on the basis of its Gram operator (the $G$-construction).