This is related to a question I recently asked on math.SE. Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the space, i.e. $\overline{\operatorname{span}}(G)=\mathcal H$.
We know that if $G$ is a frame, meaning there's $0<A\le B<\infty$ such that $$A\|x\|^2\le \sum_{k\in\mathbb{N} }\lvert \langle g_k,x\rangle\rvert^2 \le B\|x\|^2,\quad \forall x\in\mathcal H, $$ then synthesis and analysis operators are bounded, and furthermore the frame operator $S:\mathcal H\to\mathcal H$, $Sx=\sum_k g_k \langle g_k,x\rangle$, is bounded and bounded-below with bounded inverse $S^{-1}$.
More generally, if $G$ is a Bessel sequence (but still spans the space), meaning we are only ensured existence of the upper frame bound $B$, then synthesis operator $T:\ell^2(\mathbb{N})\to\mathcal H$ and analysis operator $T^*:\mathcal H\to\ell^2(\mathbb{N})$ are still well-defined and bounded (as discussed e.g. in Christensen's 2016 book, section 3.2). The frame operator is also still bounded, but it's not ensured to be bounded-below, and correspondingly its inverse won't generally be bounded.
A simple example of this would be $G=\{\frac{e_n}{n}\}_{n\in\mathbb{N} }\subset\mathcal H$. This spans the space but has no lower frame bound. The frame operator is then $S=\sum_n \frac{e_n e_n^*}{n^2}$ with inverse $S^{-1}=\sum_n n^2 e_n e_n^*$. This is unbounded as an operator in $\mathcal H$, but still makes sense as a map into the space of all sequences. For example, taking $\mathcal H=\ell^2(\mathbb{N})$, I'd get $$S^{-1} (1/n)_n = (n)_n\notin\ell^2(\mathbb{N}).$$ So I'm still seemingly getting the correct set of expansion coefficients, although the expansion coefficients are not in $\ell^2$. Then again, I have a feeling the unboundedness will give me trouble here: what if I consider $(1/n)_n+x^{(k)}$ for a sequence $x^{(k)}\in\ell^2$ such that $x^{(k)}\to0$ but $S^{-1} x^{(k)}\not\to0$? I know it exists due to the discontinuity of the operator, but wouldn't that give me a different set of expansion coefficients?
More generally, I'm interested in any theory or results on this topic, and problems arising with considering decompositions in terms of spanning Bessel sequences that are not frames.
