I would like to find a sequence $\{x_n\}\subset H$ that is an unconditional basis for a Hilbert space $H$ that is not a Riesz basis for $H$.
This is Exercise 7.15 in A Basis Theory Primer, Christopher Heil. I am just reading this book and not a homework.
Thought: Heil's book Theorem 7.13 says a sequence $\{h_n\}\subset H$ is a Riesz basis for $H$ if and only if $\{h_n\}$ is a bounded unconditional basis for $H$.
Thus, any example $\{x_n\}$ desired must be unbounded: $$ \text{Either}\quad \inf_{n}\|x_n\|_{H}=0\ \text{ or }\ \sup_{n}\|x_n\|_{H}=\infty\quad \text{must hold.} $$ Further, from the assumption for any $x\in H$ we have the unique representation $ x=\sum_{n}a_n(x)x_n $ with scalars $a_n(x)$, and the series is unconditionally convergent. This means, in view of Orlicz's Theorem, $$ \sum_{n}|a_n(x)|^2\|x_n\|_H^2<\infty. $$ So, roughly if we decide to construct $\{x_n\}$ such that $\sup_{n}\|x_n\|_{H}=\infty$, we want $x_n$ that gives coefficient functionals of small value $a_n(x)$ for any $x\in H$? But this doesn't really tell me much...
Just start with the standard basis $(e_n)$ for $\ell^2$. Now consider $(ne_n)$, which is still an unconditional basis for $\ell^2$, as each $x\in \ell^2$ can be written as $$ x = \sum_n x_n e_n = \sum_n \frac{x_n}n \cdot ne_n $$ But it is unbounded, as $\|ne_n\| = n\|e_n\| = n$.