Let $H$ be a separable, infinite dimensional Hilbert space. For an ONB $(e_n)_{n \in \mathbb{N}}$ of $H$ together with a series $(\alpha_n)_{n \in \mathbb{N}} \subset (0,\infty)$ such that $\sum\limits_{n=1}^\infty \alpha_n^2 < \infty$ and an $r>0$ we call the set \begin{align*} & E_{\alpha,e,r} := \left\{ x = \sum\limits_{n=1}^\infty \langle x,e_n \rangle_H \, e_n \ \middle| \ \sum\limits_{n=1}^{\infty} \frac{\langle x,e_n \rangle_H^2}{\alpha_n^2} \leq r \right\} \end{align*} a Schmidt-ellipsoid.
My question is if we can pack any Schmidt ellipsoid into another Schmidt-ellipsoid, which was constructed with regards to another ONB of $H$. In other words let $(e_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in \mathbb{N}}$ be two ONBs of $H$ and let $E_{\alpha,e,r}$ be as defined above. Does there exists a series $(\beta_n)_{n \in \mathbb{N}} \subset (0,\infty)$ with $\sum\limits_{n=1}^\infty \beta_n^2 < \infty$ and a $R>0$ such that $E_{\alpha,e,r} \subset E_{\beta,b,R}$?
It might be helpful to intepret the Ellipsoids $E_{\alpha,e,r}$ as Images $T(B_r)$ of certain operators $T: H\rightarrow H$, with $T(e_n) = \alpha_n \, e_n$, where $B_r := \{x \in H \mid ||x||_H^2 \leq r\}$.
