If $P_1, P_2: V \to V$ are linear projection operators on the vector space $V$ with $R := P_1(V) = P_2(V)$, is it true that any convex combination of $P_1$ and $P_2$ is again a projection operator $P_3$ with $P_3(V) = R$?
I’m trying to figure out whether or not the convex combination of two Ehresmann connections on a fiber bundle is again an Ehresmann connection, as is seemingly implied by the first paragraph of $\S$2 of this paper.
Yes, this is true.
Let $P_1, P_2:V \to V$ be projections onto $R$. I.e., $P_iP_i = P_i$ and $P_i|_R$ is the identity for $i = 1,2$. Let $c_1, c_2 \geq 0$ be such that $c_1 + c_2 = 1$. Define $P_3:= c_1 P_1 + c_2 P_2$.
It is immediate that $P_3(V) \subseteq R$. We have $$P_3P_3 = (c_1P_1+c_2P_2)(c_1P_1 + c_2 P_2) = c_1^2 P_1^2 + c_2^2 P_2^2 + c_1 c_2 P_1P_2 + c_1 c_2 P_2 P_1 = c_1^2 P_1 + c_2^2 P_2 + c_1c_2(P_1 + P_2) = c_1(c_1+c_2)P_1 + c_2(c_1+c_2)P_2 = c_1 P_1 + c_2 P_2 = P_3,$$
so $P_3$ is indeed a projection. Finally, consider any $v \in R$. Then $$P_3 v = c_1P_1 v + c_2P_2 v = c_1 v + c_2 v = v,$$
which completes the proof. Note also that, since $V = P_3(V) \oplus \ker P_3$ for any linear operator $P_3$, we have $V = R \oplus \ker P_3$.