Let $H$ be a separable Hilbert space. 


A sequence $\{f\_{n}\}$ is a frame for a separable Hilbert space $H$ if there exists $A,B>0$ such that for all $f$ in $H$
$$
A\|f\|^2 \leq \sum |\langle f, f\_n \rangle|^2 \leq B \|f\|^2
$$
$\{f\_{n}\}$ is complete if the only element which is orthogonal to all $f\_{n}$ is the zero element.


It is known that if $\{f\_{n}\}$ is a frame then it is complete, but the converse is not true. In which cases the converse will be true, i.e. 


When does a complete sequence $\{f\_{n}\}$ becomes a frame for $H$ (or at least satisfying the lower frame bound)?


Of course the converse is true if $f\_n$ is orthonormal. But orthogonality is a very strong condition for me.


Any comments or references are welcome!



Frames and completeness
Presumably you'd like some answer less tautological that "a complete sequence is a frame when it satisfies the defining condition for frames," but then it isn't clear what your rules are.


Without loss of generality, you could take $\ell^{2}$ for your $H$. Then using your $f\_n$'s as rows, a sequence takes the form of a matrix (with rows in $\ell^2$). A priori, such a matrix defines an operator from $H$ to ${\Bbb C}^{\Bbb N}$. Complete means kernel $\{0\}$; frame means bounded operator, with spectrum bounded away from 0, to $\ell^2$. So I read your question as asking for a characterization of boundedness and/or spectrum bounded away from 0, directly from the appearance of the matrix coefficients. I don't believe that question admits any satisfactory general answer. 


