I have been stuck on this one part of this question for about 3 weeks now and I just have no ideas. I have a Hilbert space $H$ and another Hilbert space $l^2$ which is the space of sequences for which $\sum\limits_{1}^{\infty}\|a_j\|<\infty$. I also have the definition of a frame and that there are constants $A$ and $B$ with $0<A<B<\infty$ My frame is called $\{e_j\}_{j=1}^{\infty}$. The frame definition I am given is $A\|x\|^2\leq\sum\limits_{1}^{\infty}|\langle x,e_j \rangle|^2\leq B\|x\|^2$
Next I have operators $D(x)_j=\langle x,e_j\rangle$, $S$, the adjoint of $D$ and $T=SD$
Now, I have found the action of $T$ and $S$, and importantly I have $Tx=\sum\limits_{j=1}^{\infty}\langle x,e_j\rangle e_j$ I also found that T is self adjoint and invertible. Then I found an expansion for $x$ of $x=\sum\limits_{j=1}^{\infty}\langle x,f_j \rangle e_j$, with $f_j=T^{-1}e_j$
Now after all the context my actual question that I have issues with is that I need to show that $\|I-\frac{2}{A+B}T\|\leq \frac{B-A}{B+A}<1$
I noticed that $\|Tx\|=\sum\limits_{1}^{\infty}|\langle x,e_j \rangle|^2$ and so I think I can say that, $A\leq\|T\|\leq B$ and I was trying to work from there and the closest I got was to $\|\|I\|-\frac{2}{A+B}\|T\|\|\leq \frac{2A-2B}{A+B}$ But I really just am not sure how to get any closer so any help would be appreciated.
Thanks.