I'm reading the proof of the Cotlar–Stein almost-orthogonality lemma from Wikipedia page. I have several questions concerning the proof. We use the notation from the Wikipedia page.
Why do the partial sums ${s_{n}=\sum _{i=1}^{n}T_{i}v}$ form a Cauchy sequence?
The key estimate given there is: If $R_1,\ldots,R_n$ is a finite collection of bounded operators, then
$${\displaystyle \displaystyle {\sum _{i,j}|(R_{i}v,R_{j}v)|\leq \left(\max _{i}\sum _{j}\|R_{i}^{*}R_{j}\|^{1 \over 2}\right)\left(\max _{i}\sum _{j}\|R_{i}R_{j}^{*}\|^{1 \over 2}\right)\|v\|^{2}.}}$$
We have $$\Bigg\|\sum _{i=m}^{n}T_{i}v\Bigg\|^{2}\leq \sum _{i,j\geq m}|(T_{i}v,T_{j}v)|\leq\bigg(\sup_{i\geq m}\sum_{j=m}^\infty\|T_{i}^{*}T_{j}\|^{\frac{1}{2}}\bigg)\bigg(\sup_{i\geq m}\sum_{j=m}^\infty\|T_{i}T_{j}^*\|^{\frac{1}{2}}\bigg)\|v\|^2.$$
However, I don't think the right-hand side converges to $0$ when $m\to\infty$. It is only given that $${\displaystyle A=\sup _{j}\sum _{k}{\sqrt {\|T_{i}^{*}T_{j}\|}}<\infty ,\qquad B=\sup _{j}\sum _{k}{\sqrt {\|T_{i}T_{j}^*\|}}<\infty.}$$ It is entirely possible that, for example, $$\sup_{i\geq m}\sum_{j=m}^\infty\|T_{i}^{*}T_{j}\|^{\frac{1}{2}}$$ stays positive and constant, e.g., take any convergent series $\sum_{i=-\infty}^\infty c_i$ of positive reals and set $\|T_i^*T_j\|=c_{j-i}^2$. So I think there must be something that restricts these quantities, or I'm in the wrong path.
Any help will be appreciated!
Best Answer
I think the argument works as follows:
First we state that under the hypotheses of the lemma we have
$$ \sum_{i,j = 1}^\infty |(T_i v, T_j v)| \leq AB \|v\|^2. $$ As $|(T_i v, T_j v)| > 0$, this gives us absolute convergence of the series $\sum_{i,j} |(T_i v, T_j v)|$.
In particular $\sum_{i,j \geq m} |(T_i v, T_j v)| \to 0$ for $m \to \infty$ (necessary for convergence).
To conclude that the sequence $\sum_{i = 1}^n T_i v$ is Cauchy we deploy the estimate $$ \| \sum_{i = m}^n T_i v\|^2 \leq \sum_{i,j \geq m} |(T_i v, T_j v) |. $$
Completeness gives us that the limit $\lim_{n \to \infty} \sum_{i = 1}^n T_i v$ exists.
This allows us to estimate $$ \| \sum_{i = 1}^\infty T_i v \|^2 = \lim_{n \to \infty} \|\sum_{i = 1}^n T_i v \|^2 \leq AB \|v\|^2. $$