Sequence of unit vectors in a Hilbert space

Question

Question: Let $H$ be a Hilbert space and $\{\xi_{i}\}\subset H$ be a sequence of unit vectors. Suppose that $||T_{j}(\xi_{i})-\xi_{i}||\rightarrow0$ as $i\rightarrow\infty$, for $j=1, 2, ...n$ (here $T_{j}\in B(H)$). Then can we get that $$\frac{1}{n}||\sum\limits_{j=1}^{n}T_{j}(\xi_{i})||\geq 1?$$

Answer 1

Let $T_j=T:\ell^2\to \ell^2$ for all $j$ where $Tx=x$ and let $\xi _i=e_i$ in $\ell^2$. Then $\|\sum_{j=1}^{n} Te_j\|_2=\sqrt {n}$ and $\frac {\sqrt {n}}{n}< 1$.

Answer 2

The statement in your question $$ \frac{1}{n}||\sum\limits_{j=1}^{n}T_{j}(\xi_{i})||\geq 1 $$ could be true for all $n$ or could even be false for all $n$. The hypotheses you state are not enough to determine.

• Mitsos' answer gives an example of this being false for all $n$.

• Let $H = \mathbb{R}$, $\xi_i = 1$, and $T_j(xi_i) = 1 + \frac{1}{n}$. Then $\displaystyle \sum_{j=1}^n T_j(\xi_i) > n$, so your statement is true for all $n$.

The presence of linear operators in your question is a red herring, because $T_j(\xi_i)$ can essentially be any elements of $H$ you feel like. This seems to be implied by this version of the Hahn-Banach theorem.