I ran into the following question about bounded operators on Hilbert spaces; I could really use some help.
It goes like this: Suppose that $\left\{T_{k}\right\}$ is a collection of bounded operators on a Hilbert space $\mathcal{H}$, with $||T_{k}|| \leq 1$ for all $k$. Suppose also that $T_{k}T_{j}^{*}=T_{k}^{*}T_{j}=0$ for all $k \neq j$. Let $S_{N}=\sum_{k=-N}^{N}{T_{k}}$. Then, we have that $S_{N}(f)$ converges as $N \rightarrow \infty$, for every $f \in \mathcal{H}$. Moreover, if $T(f)$ denotes the limit, then $||T|| \leq 1$.
What if the condition $T_{k}T_{j}^{*}=T_{k}^{*}T_{j}=0$ is replaced by $||T_{k}T_{j}^{*}|| \leq a_{k-j}^{2}$ and $||T_{k}^{*}T_{j}|| \leq a_{k-j}^{2}$ for positive constants $a_{n}$ with the property \sum_{-\infty}^{\infty}{a_{n}}=A < \infty? Is is true that the convergence preserves and $||T|| \leq A$?