On connection between the norm of a sum of orthoprojections onto subspaces and the norm of the product of orthoprojections onto the corresponding orthogonal complements

Abstract

We show that if $\epsilon I \le P_1+P_2+\dots+P_k $, where $I$ is the identity operator and $P_1,\dots, P_k$ are orthoprojections that project onto linearly independent subspaces, then the norm $\|P_1+P_2+\dots+P_k\|\le k-(k-1)\epsilon$. It was found that without the linearly independent condition there exit orthoprojections such that $\|(I-P_k)(I-P_{k-1})\dots(I-P_1)\|\ge 1-17\epsilon/k^2$.