Let H be a Hilbert space .Is there always a non orthogonal Riesz basis $D$ on it such that following holds?
$\sup_{g\in D }\sum_{g'\in D,g'\not=g}|\langle g,g'\rangle|<1/3 $
And is there Riesz such that the inequality does not hold?
Let H be a Hilbert space .Is there always a non orthogonal Riesz basis $D$ on it such that following holds?
$\sup_{g\in D }\sum_{g'\in D,g'\not=g}|\langle g,g'\rangle|<1/3 $
And is there Riesz such that the inequality does not hold?
Fix an orthonormal basis $\{e_j\}_{j\in J}$. For each $j\in J$ we denote by $j+1$ some fixed element different from $j$ (it can actually be $j+1$ if $J=\mathbb N$).
Define $ f_j=e_j+\frac18\,e_{j+1},\ \ j\in J. $ The set $\{f_j\}$ clearly spans $H$. Also, $ \left\|\sum_jc_jf_j\right\|^2=\sum_{j,k}\langle c_je_j+\frac{c_j}8e_{j+1},c_ke_k+\frac{c_k}8e_{k+1}\rangle=\sum_j|c_j|^2+\frac18\,\sum_j|c_j|^2+2\text{Re}\,\frac18\,\sum_jc_j\overline{c_{j+1}}. $ Note that, by Cauchy-Schwarz, $|\sum_jc_j\overline{c_{j+1}}|\leq\sum_j|c_j|^2$. Then $ \left\|\sum_jc_jf_j\right\|^2\leq\frac98\sum_j|c_j|^2+\frac14\,\sum_j|c_j|^2=\frac{11}8\sum_j|c_j|^2. $ Also, $ \left\|\sum_jc_jf_j\right\|^2\geq\frac98\sum_j|c_j|^2-\frac14\,\sum_j|c_j|^2=\frac{7}8\sum_j|c_j|^2. $ All this shows that $\{f_j\}$ is a Riesz basis.
Finally, for any $j\in J$, $ \sum_{k\ne j}|\langle f_k,f_j\rangle|=\frac18|\langle f_{j+1},f_j\rangle|+\frac18|\langle f_j,f_{j+1}\rangle|=\frac18+\frac18<\frac13. $