This question was asked in my assignment of Functional Analysis and I am not able to make progress in few parts.
Question: Let H be an Hilbert space and let $(f_i)_{i\in I}$ be a (finite or infinite) sequences of elements in H. We say that $f_i$ is a bessel sequence if there exists a constant $C< \infty$ such that for every $x\in H$ : $\sum_{i\in I} |<x, f_i> |^2 \leq C ||x||^2$; and we say that $(f_i)$ is frame if there exists 2 constants $C< \infty$ and c >0 as for every $x\in H: c||x||^2 \leq \sum_{i\in I} |<x,f_i>|^2 \leq C||x||^2$.
(1) Show that $(f_i)$ is a bessel sequence iff there exists a continuous linear operator $T: l^2(I) \to H$ st $Te_i =f_i$ for every $i\in I$, where $(e_i)_{i\in I}$ is the canonical basis of $l^2$. Also, identify the adjoint of T.
(2) Show that $(f_i)$ is a frame iff there is a surjective operator R and $T: l^2(I) \to H$ such that $Te_i =f_i$ for every $i\in I$.
Attempt:(1) Assuming the existence of continuous operator I have proved that it is bessel but I am not able to prove converse. Also , I am not able to find the adjoint.
(2) I am not able to prove the existence of surjective operator. Conversaly, using the existence of surjective operator I have deduced $ \sum_{i\in I} |<x,f_i>|^2 \leq C||x||^2$ but I am not able to deduce a lower bound.
Kindly help me with this.
Thanks!
For question $1$: Suppose there exists a continuous linear operator $T$ such that $T(e_i) = f_i$. You want to show that $f_i$ is Bessel meaning that $\sum_{i \in I} |\langle x, f_i\rangle|^2 \leq C||x||^2 $. Using our assumption we have
\begin{align*} \sum_{i \in I} |\langle x, f_i\rangle|^2 & = \sum_{i \in I} |\langle x, T(e_i)\rangle|^2\\ &= \sum_{i \in I} |\langle T^*x, e_i\rangle|^2 \\ & = ||T^*x||^2\\ & \leq ||T||^2 ||x||^2 \end{align*}
To find $T^*$ simply go backwards in this chain of equalities and "make it true". In other words consider $S(x)$ to be the $l^2$ sequence whose $i^{th}$ component is $\langle x, f_i \rangle$. That is
$$S(x) := \{\langle x, f_i \rangle \}_{i=0}^{\infty} $$
Then $\langle S(x) , e_i \rangle = \langle x, f_i \rangle$. Then we have $\langle x,T(e_i) \rangle = \langle Sx, e_i \rangle $ for all $x \in \mathcal{H}$. By linearity (you should be able to fill in the details here), we have that that $ \langle x,T(y) \rangle = \langle Sx, y \rangle $ for all $x,y \in \mathcal{H}$.