$\mathcal H$ being a Hilbert space, $\{g_k\}_{k \in N}$ is a Bessel sequence if there exsits $B >0$ such that $\forall f \in \mathcal H$, $\sum_{k\in N} |\langle f,g_k\rangle|^2 \leq B \| f \|^2$. And it is a frame if there exists also $A >0$ such that $A \| f \|^2 \leq \sum_{k\in N} |\langle f,g_k\rangle|^2$. Now, if $A \| f \|^2 \leq \sum_{k\in N} |\langle f,g_k\rangle|^2\leq B \| f \|^2$ holds only for $f \in \overline{Span}(\{g_k\}_{k\in N})$, then $\{g_k\}_{k \in N}$ is said to be a frame sequence.
My question is : if $\{g_k\}_{k \in N}$ is a Bessel sequence, is it also a frame sequence ? On other words, is a Bessel sequence a frame for its closed linear span ? And, conversely, if $\{g_k\}_{k \in N}$ is a frame for its closed linear span, is it a Bessel sequence for entire $\mathcal H$ ?
I have finally found the answers to both questions :
1) if $\{g_k\}_{k\in N}$ is a Bessel sequence for its closed linear span, is it a Bessel sequence for entire $\mathcal{H}$ ? The answer is yes. Indeed, suppose $\{g_k\}_{k \in N}$ is a Bessel sequence for its closed linear span with Bessel bound $B$, and let $h \in \mathcal H$. Then, $\exists h_1,h_2$ such that $h=h_1+h_2$ with $h_1 \in \overline{span}(\{g_k\}_{k\in N})$ and $h_2 \in \overline{span}(\{g_k\}_{k\in N})^\perp$. Then, $\langle h,g_k \rangle = \langle h_1,g_k\rangle$ and $\sum_{k\in N} |\langle h,g_k \rangle|^2 = \sum_{k\in N} |\langle h_1,g_k \rangle|^2 \leq B \|h_1\|^2 \leq B \|h\|^2$.
2) if $\{g_k\}_{k\in N}$ is complete in $\mathcal H$, does $\{g_k\}_{k\in N}$ admits a lower bound ? The answer is no. Consider $\{e_k\}_{k\geq 1}$ an orthonormal basis of $\mathcal H$ and $f_k = e_k/k$, then $\{f_k\}_{k\geq 1}$ is obviously complete but $\sum_{k\geq 1} |\langle e_n,f_k \rangle|^2 = 1/n$ which tends to 0 when $n$ tends to infinity. Therefore, generally, if $\{g_k\}_{k \in N}$ is a Bessel sequence, it may not be a frame sequence.