Does the sum of infinitely many orthogonal projections converge with respect to the operator norm?

Question

Let $H$ be a Hilbert space and let $E\subset H$ be an orthonormal set in $H$. For $F\subset E$, let $P_{F}\colon H\to H$ denote the orthogonal projection on the closed linear span $\bigvee F$ of $F$. Now let $F_{1},F_{2},\ldots\subset E$ be pairwise disjoint subsets of $E$. I have shown that for every $h\in H$, we have $$\sum_{k\geq0}P_{F_{k}}h=P_{\bigcup_{k\geq0}}h.$$ Now I want to say something about the convergence (or non-convergence) of $\sum_{k\geq0}P_{F_{k}}$ in $\mathscr{B}(H)$ with respect to the operator norm. Sadly enough, I can't even think of an example that satisfies the conditions of this setting. Any suggestions to get me started would be greatly appreciated.

Answer 1

Not true. Let $(e_n)$ be an orthonormal basis for $H$. Let $F_k=\{e_k\}$. Then $P_{F_k}x= \langle x, e_k \rangle e_k$ and the series $\sum \langle x, e_k \rangle e_k$ does not converge in operator norm becasue the operator norm of the general term is $1$: $\sup \{\|\langle x, e_k \rangle e_k\|: \|x\| \leq 1\} =1$ for each $k$.