Let $H$ be a Hilbert space and let $\{x_n\}_{n\in\mathbb N}$ and $\{y_n\}_{n\in\mathbb N}$ be orthonormal sets (not necessarily maximal). Consider the operators $T_n$ defined by
$$T_nu=(x_n|u)y_n.$$
Does the series $\sum_{n=1}^\infty T_n$ converge uniformly? I.e., does the sequence of partial sums $S_n=\sum_{i=1}^n T_n$ converge in the operator norm of $\mathfrak B(H)$?
I'm asking because the book I am reading (V. Moretti, Spectral Theory and Quantum Mechanics) casually mentions that if such a series converges uniformly, then the limit is a compact operator [Example 4.16 (2)]. But I am curious whether such a series must always converge uniformly anyway.
Let $k, n$ be natural and $||u|| = 1$. We have $$\left|\left|\sum_{m=n}^{k+n}(x_m, u)y_n\right|\right|^2 = \sum_{m=n}^{k+n}|(x_m,u)|^2 \to 0,$$ when $n \to \infty$, because we can extend $\{x_n\}$ to a basis. But this sum has a maximal value $1$ (by the Bessel inequality) for any $n$ and $k$. Moreover, when $u \in \mathop{span}(x_n, \dots, x_{n+k})$ this value is achieved. In other words, $||S_{n+k} - S_{n}|| = 1$ for all $n, k \in \mathbb{N}$.