TITLE: Projections in Banach spaces
QUESTION [10 upvotes]: Dear All, 
I am absolutely lost in the following problem:
Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm min}(s,t)}$. Let $Q$ be a bounded linear operator on $X$ such that $QP_s = P_sQ$ for every $s \in [0,1]$ and the function $$s \mapsto P_s Q $$ is continuous in operator norm. Does it follow that the function must be constant (i.e. $P_sQ \equiv P_0Q$)? 
For simplicity, one can also assume that the function $s \mapsto P_s$ is strongly continuous. I can give an affirmative answer only in some trivial situations (finite dimensional case, Hilbert spaces with family of orthogonal projections) but nothing more.

REPLY [8 votes]: Here a simple example :
Let X be the cartesian product of $L^{\infty}$ and $L^{1}$ on the
interval $[0,1]$, let $P_{t}$ the canonical projection on the subspace
of functions with support $[0,t]$ and choose $Q(f_{1},f_{2}) =
(0,f_{1})$ .