# $\epsilon$-nearly isoclinic

Question: Two $k$-dimensional subspaces $W_1,W_2$ with associated orthogonal projections $P_1, P_2$ are isoclinic with parameter $\lambda \ge 0$ if $P_1P_2P_1=\lambda P_1$ and $P_2P_1P_2=\lambda P_2$.

I was able to show that if $W_1\perp W_2$, then $W_1,W_2$ are isoclinic with $\lambda=0$. Now I've read the following definitions in a paper.

• Two subspaces $W_1,W_2$ are $\epsilon$-nearly orthogonal if for all unit vectors $\phi\in W_1$ and $\psi\in W_2$ we have $|\langle\phi,\psi\rangle|<\epsilon$.
• Two k-dimensional subspaces $W_1,W_2$ are $\epsilon$-nearly isoclinic if there exists $\lambda\geq0$ such that $(\lambda-\epsilon^2)P_1\le P_1P_2P_1\le(\lambda+\epsilon^2)P_1$ and $(\lambda-\epsilon^2)P_2\le P_2P_1P_2\le(\lambda+\epsilon^2)P_2$.

Then the claim is again, if $W_1,W_2$ have same dimension, then $\epsilon$-nearly orthogonal implies $\epsilon$-nearly isoclinic with parameter $\lambda=0$. In the paper they treat it like it is trivial. But for me, regrettably, it is not. Is there really a short proof for that or does anybody know where i can find a proof for that (if it is not so short)?

• Phrase there exists $\lambda=0\$ puzzles me. Commented Jan 23, 2015 at 0:07
• How do you define an inequality $\ P\le Q\$ for two linear operators? (I have a candidate but would like to be sure). Also, does space mean Euclidean space (I am sure that it does but it'd be nice to say so explicitly). Commented Jan 23, 2015 at 0:15
• Should there be norms in your second condition--then inequalities would make sense without any additional (exotic :-) definitions.. Commented Jan 23, 2015 at 0:19
• Ah sorry, I did not mention that. $Q\geq 0$ is defined by $\langle Q\varphi, \varphi\rangle\geq 0$ for all $\varphi\in\mathcal{H}$ where $\mathcal{H}$ is a Hilbert space. Commented Jan 28, 2015 at 21:59
• So $W_1$ and $W_2$ are subspaces of an arbitrary Hilbert space, not necessarily an Euclidien space. Also other scalar products are allowed. Greetings =) Commented Jan 28, 2015 at 22:12

Let $a$ be an arbitrary Element of a Hilbert space.
$\Vert P_2 P_1 a\Vert^2=\langle P_2 P_1a,P_1a\rangle=\Vert P_2 P_1 a\Vert\Vert P_1a\Vert\langle \frac{P_2P_1a}{\Vert P_2 P_1 a\Vert},\frac{P_1a}{\Vert P_1 a\Vert}\rangle<\epsilon\Vert P_2 P_1 a\Vert\Vert P_1a\Vert$.
$\implies \Vert P_2 P_1 a\Vert<\epsilon\Vert P_1a\Vert$
$\implies\Vert P_2 P_1 a\Vert^2<\epsilon^2\Vert P_1a\Vert^2$
$\implies\langle P_1P_2P_1a,a\rangle\le\epsilon^2\langle P_1a,a\rangle$.
$-\epsilon\Vert P_1a\Vert^2<-\Vert P_2 P_1a\Vert^2<\Vert P_2 P_1a\Vert^2$,
$-\epsilon^2\langle P_1a,a\rangle\le\langle P_1P_2P_1a,a\rangle$.
Changing the roles of $P_1$ and $P_2$ yields the claim.