$T:\ell^2 \to \ell^2$ is a compact operator

Question

Any example in Kreyszig: Introductory Functional Analysis with Applications:

Prove compactness of $T:\ell^2 \to \ell^2$ defined by $y=(\eta_j)=Tx$ where $\eta_k=\xi_j/j$ for $j=1,2,\dots$.

Defined first is $T_n x= (\xi_1, \frac{\xi_2}{2}, \frac{\xi_3}{3}, \cdots, \frac{\xi_n}{n}, 0,0, \cdots )$

Then this is compact because of 8.1-4(a): "If $T$ is bounded and $\dim T(X)<\infty$ the operator $T$ is compact."

What is the purpose of this preliminary $T_n$, then the proof goes to show that $T_n\to T$ and hence $T$ is compact. Is the fact that at the nth point here we have zeros from then onward implying that $\dim T(X)<\infty$?

This is due to theorem 8.1-5. "Let $(T_n)$ be a sequence of compact linear operators from a normed space $X$ into a Banach space $Y$. If $(T_n)$ is uniformly operator convergent, say, $\lVert T_n-T\rVert \to 0$, then the limit operator $T$ is compact."

Is this the standard way to prove operators that have $\dim T(X)=\infty$? Will I be able to extend this type of proof for $T:L^2([a,b])\to L^2([a,b])$ with polynomials?

Answer 1

Purpose of $T_n$

The purpose of considering this sequence of operators is in fact to pave the way for applying the theorem 8.1-5 from the second quote. The zeros from $n$ onward imply that $\dim{(T_n) }<\infty$ rather than $\dim{(T_n) }<\infty$, so by 8.1-4(a) $T_n$ is compact. These two little steps give you a sequence of compact operators.

I can not provide a formal way of coming up with the sequence $(T_n)_n$, but sequences like this frequently appear as examples in courses and literature, so after a while you might develop some kind of instinct for this type of thing.

Throw it all together: $T_n \rightarrow T$

The rest is straightforward since you have theorem 8.1-5 at hand. You have a sequence of compact operators of which you know the limit, which is in fact $T$.

Is this the standard way?

I'd say that there are special cases where this strategy works well, which probably look quiet similar to this one. But in my (very limited) experience, the theory of compact operators has much more to offer than this strategy. I recommend looking applications of the Arzela-Ascoli theorem.