"Uniformly" Cauchy series of operators on $L_p$ spaces. Let $(X_j, \mu_j)_{j=0}^\infty$ be a countable family of measure spaces and let $p \in (1,\infty)$.
Consider a sequence $(T_j)_{j=1}^\infty$ of bounded linear operators $T_j: L_p(X_0) \to L_p(X_j)$ such that
$$
\sup_{\|f\|_p=1} \left( \sum_{j=1}^\infty \| T_jf\|_p^p \right)^{1/p} < \infty.
$$
Question: Let $\epsilon >0$. Is it true that there is $N \in \Bbb{N}$ such that for all $m>n>N$
$$
\sup_{\|f\|_p=1} \left( \sum_{j=n}^m \| T_jf\|_p^p \right)^{1/p} < \epsilon?
$$
Progress: I think I've proved this for $p=2$. Indeed, in this case we use adjoints and notice that $$\sup_{\|f\|_2=1} \left( \sum_j\| T_jf\|_2^2 \right)^{1/2}=\left\|\sum_{j}T_j^*T_j \right\|,$$ whence the desired property follows from the convergence of the series $\sum_j T_j^*T_j$ in $\mathcal{B}(L_p(X_0))$. However, I do not quite see how to approach this for a different value of $p$.
 A: This is not true (even for $p=2$). Take $X_0= \mathbb N$ with counting measure, $X_j=\{j\}$,
and $T_j(f) = f_j$, i.e, $T_jf$ is the $j$-entry of $f\in l^p$.
Then
$$
\sum_j \|T_jf\|_p^p= \|f\|_{l^p}
$$
but
$$
\sup_{\|f\|_{l^p}\le 1} \sum_{j=n}^m  \|T_jf\|_p^p=1
$$
with the choice $f=e_n$.
The problem with your proof is that $\sum_{j=1}^\infty T_j^*T_j$ does not converge in the operator norm.
A: Here is a counter example in the case where $X_j=X$ for all $j$.
Consider the example $(X,\mathscr{F},\mu)=((0,\infty),\mathscr{B}((0,\infty)),\lambda)$ ($\lambda$ is Lebesgue measure).
Define the operators
$T_nf=f\mathbb{1}_{(n-1,n]}$.
Then
$$\Big(\sum_n\|T_nf\|^p_p\Big)^{1/p}=\|f\|_p.$$
Furthermore, for any $m\leq n$,
$$\sum^n_{j=m}\|T_jf\|^p_p=\int^n_{m-1}|f|^p\,d\lambda.$$
We can consider any $f\in L_1((0,\infty),\lambda)$ and a a function defined over all $\mathbb{R}$ by setting $f(x)=0$ for $x\leq0$.
Fix $f\in L_p$ with $\|f\|_p=1$ and define $f_N=f(\cdot-N)$, (translate the origin to $N$).
Notice that $\|f_N\|_p=\|f\|_p$ for all $N$.
Choose $M$ large enough so that
$$\frac{1}{2}\leq\int^M_0|f|^p\,d\lambda.$$
For any $N$, $$\sum^{N+M}_{j=N+1}\|T_jf_N\|^p_P=\int^{N+M}_{N}|f_N|^p\,d\lambda=\int^M_0 |f|^p\,d\lambda\geq\frac12.$$
