so I am reading Krengel's text on Ergodic theorems.
And the next lemma bugs me as for the proof of it.
It's by Foguel and Weiss.
Statement:
If $P_1, P_2$ are commuting elements of a Banach algebra with $||P_1||=||P_2|| = 1$, and $Q= \alpha P_1 + (1-\alpha)P_2$ $(0<\alpha < 1)$, then: $||Q^n (P_1-P_2)|| \leq K\alpha (1-\alpha) n^{-0.5}$ for some constant $K$.
I thought that induction will do, but the basis of induction doesn't follow, I mean induction on $n$.
First we prove the theorem for the case when $\alpha=1/2$ and $\beta=1/2$. In this case, we have $$Q^n (P_1-P_2)=\frac{1}{2^n}\sum_{k=0}^n\binom{n}{k}P_1^kP_2^{n-k}(P_1-P_2)\\=\frac{1}{2^n}\sum_{k=1}^n[\binom{n}{k-1}-\binom{n}{k}]P_1^kP_2^{n-k-1}+\frac{1}{2^n}P_1^{n+1}-\frac{1}{2^n}P_2^{n+1}$$ thus we have that $$||Q^n (P_1-P_2)|| \leq \frac{1}{2^{n-1}}+\frac{1}{2^n}\sum_{k=1}^n|\binom{n}{k-1}-\binom{n}{k}|.$$
We assume that $n$ is even. For $n$ odd it is similar. Since $\binom{n}{k}$ increases as $k$ increases from $0$ to $n/2$ and then decreases as $k$ goes from $n/2$ to $n$ the sum of absolute values is bounded by $(2/2^n)\binom{n}{n/2}$ which is, by Stirling's formula, bounded by $K/4\sqrt{N}$ for some $K$.
For the general case, the equality seems not be true. For example, let the Banach algebra to be the space of bounded linear operators of a Hilbert space $\mathcal H$, and $P_1$ be $P_2$ be two projection to two closed subspaces with intersection $\{0\}$. For the case $n=1$, we see that $$Q(P_1-P_2)=\alpha P_1- (1-\alpha)P_2$$ and the norm of it is $\sqrt{1-2\alpha+\alpha^2}$ which can not be bounded by a constant multiplied by $\alpha(1-\alpha)$.