This is a homework problem that I'm having some trouble with, so any hints would be appreciated. Let $H$ be a Hilbert space with an orthonormal basis $\{e_n\}$. Consider the operator $F : H\rightarrow H$ defined by $$Fx = \sum_{n=1}^\infty \beta_n \langle x, e_n\rangle e_n,$$
where $\{\beta_n\}\subset \mathbb C$
Suppose that $F$ is compact. (That is, the image of any bounded sequence contains a convergent subsequence). Show that $\lim_{n\rightarrow\infty} \beta_n = 0$.
The obvious approach is to choose a particular bounded sequence and try and get that to give me the result, and the first sequence I tried was obviously $\{e_n\}$. So the image of this sequence is $\{\beta_ne_n\}$, so this implies that there is a subsequence $\{\beta_{n_j}e_{n_j}\}$ which converges. But this doesn't help me and I can't see how else to approach the question.
Best Answer
I like your idea of building a sequence from the orthonormal basis vectors and proving non-existence of convergent subsequences! But I think we need to be a bit more restrictive in how we select our sequence: rather than selecting all the orthonormal basis vectors to be our sequence, we should select only the "troublesome" ones.
So here is the idea. Suppose, for contradiction, that $\beta_n$ does NOT tend to zero. Then hopefully you can show that there exists a $\epsilon > 0$ and there exists an ascending sequence $n_1, n_2, n_3, \dots$ such that $$ |\beta_{n_1}| > \epsilon, \ \ \ | \beta_{n_2} | > \epsilon, \ \ \ | \beta_{n_3} | > \epsilon \dots $$
Now consider the sequence $$ e_{n_1}, \ \ e_{n_2}, \ \ e_{n_3}, \ \ \dots$$ Hopefully you can verify that the sequence $$F(e_{n_1}), F(e_{n_2}), F(e_{n_3}), \dots$$ cannot possibly contain a Cauchy subsequence, which would be enough to complete your proof.
To see the problem with using the entire orthonormal basis $e_1, e_2, e_3, \dots$ as the "test" sequence in your argument: Consider the example where $\beta_n$ is $1$ when $n$ is odd and $0$ when $n$ is even. Then $F(e_n)$ clearly contains a convergent subsequence, namely, $F(e_2), F(e_2), F(e_6), \dots$ So we don't get a contradiction. We need to be more restrictive, by picking only the "troublesome" elements $e_1, e_3, e_5, \dots$ to be our "test" sequence, and only then do we find ourselves unable to find a Cauchy subsequence.