Let $x,y$ be vectors of some Hilbert space of unit length.

Then we can consider the projection $P_x:=\langle \bullet, x \rangle x$ and similarly $P_y.$

Assume then that we know that $\left\lVert x-y \right\rVert\le \alpha.$

What is the optimal estimate on $\left\lVert P_x-P_y \right\rVert_1$ we can get, where $1$ denote the trace norm.

I am very grateful for any comment/remark on this.

A very pedestrian estimate would be to smuggle in the map $P_{xy}=\langle \bullet,x \rangle y.$ This way we can use the triangle inequality to estimate

$\left\lVert P_x-P_y \right\rVert_1 \le 2 \left\lVert x-y \right\rVert$

which does not seem to be very sophisticated. Is there a better estimate?-I cannot see when my trivial estimate is optimal.