Let $\mathcal{B}(\ell^p)$ denote the space of bounded linear operator on $\ell^p$ and let $T, T^{-1} \in \mathcal{B}(\ell^p)$ with $\|T\|=1$. I'm trying to understand what is the closed subalgebra generated by $\{T,T^{-1}\}$ (let's call it $A$).
Surely $A$ contains operators of the form $\sum\limits_{k=-n}^{n}a_k T^k$. Since $A$ is closed, it must contains some operators of the form $\sum\limits_{k=-\infty}^{\infty}a_k T^k$. What conditions should $\{a_k\}_{k= -\infty}^\infty$ satisfy so that $\sum\limits_{k=-\infty}^{\infty}a_k T^k$
1) converges ?
2) belongs to $\mathcal{B}(\ell^p)$ ?
My educated guess : for both 1) and 2) it suffices that $\{a_k\}_{k=-\infty}^\infty$ belong to $\ell_1$ (but is it a necessary condition ?).
Let $\xi \in \ell_p$ with $\|\xi\|=1$, $$\left\|\left(\sum\limits_{k=-\infty}^{\infty}a_k T^k\right)(\xi)\right\|_p \leq \sum\limits_{k=-\infty}^\infty |a_k|\cdot \|T(\xi)\|^k \leq \sum\limits_{k=-\infty}^\infty |a_k|.$$
This is far from a complete answer (I don't think that a full characterization is at all possible, though I might be wrong).
The two questions are the same. The convergence of the series is the convergence (in $\mathcal B(\ell^p)$) of the sequence of partial sums. As $\mathcal B(\ell^p)$ is complete, if the series converges, it has to converge to a bounded operator. The story could be different if we consider convergence of the series in a topology in which $\mathcal B(\ell^p)$ is not complete.
The condition $a\in\ell^1$ is not necessary in general. For example, if $T=-I$ and $a_n=1/n$, the series converges even though the coefficients are not in $\ell^1$.
Even when $p=2$ the situation is not clear to me. If $T$ is selfadjoint, then the algebra you want is $C(\sigma(T))$, because it is a C$^*$-algebra. But when $T$ is not selfadjoint, it is not obvious what you get.