$\underset{n\rightarrow\infty }{\lim}\Vert \underset{i\geq 0}{\sum }\lambda _{i}^{n}x_{i}\Vert^{\frac{1}{n}}\not=0$ if $\lambda _{i}\not=0$?

Question

Let $H$ be a Hilbert space. Suppose we have a basis $\left( x_{i}\right) _{i\geq 0}$ for $H$ (not necessarily orthonormal) and a sequence $\left( \lambda _{i}\right) _{i\geq 0}$ in $\mathbb{C}$, such that $\underset{i\geq 0}{\sum }x_{i}$ converges to a non-zero vector, $\underset{% n\rightarrow \infty }{\lim }\lambda _{i}=0$ and $\lambda _{i}\not=0$ for all $i\geq 0$. Do we have $\underset{n\rightarrow \infty }{\lim }\Vert \underset{i\geq 0}{\sum }\lambda _{i}^{n}x_{i}\Vert ^{\frac{1}{n}% }\not=0$ ? If not, what additionally condition we should make on $\left( \lambda _{i}\right) _{i\geq 0}$ to have this conclusion. I am not familiar with this type of problem when the basis is not orthogonal, and I think that the answer would be very complex. To be more precise, I am facing this problem for a Riesz operator when $\left( x_{i}\right) _{i\geq 0}$ represents a sequence of eigen vectors and $\left( \lambda _{i}\right) _{i\geq 0}$ a sequence of corresponding non-zero eigen values.