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I'm trying to study compact operators, but i'm having a little trouble with the 'practice'.. What are some tecniques to prove an operator compact. I know it can be shown that a limit of finite range operators is a compact one, but other ways? For instance take $T_{\alpha}:C[0,1] \longrightarrow C[0,1]$, $$T_{\alpha}f(x)=\int_0^x\frac{f(t)}{t^{\alpha}} dt$$ for $\alpha \in [0,1[$. How to prove it is compact and what is its spectrum?

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I haven't thought about your specific case, but Arzela-Ascoli is a standard tool for showing subsets of $C[0,1]$ are compact. – Chris Eagle Apr 15 '12 at 22:09
I took a bounded sequence of functions $f_n$ and showed that $Tf_n$ is equicontinuos and uniformly bounded, and this should give the proof of compact. What about the spectrum? – balestrav Apr 26 '12 at 13:45

Some methods that I know are

1)T(B(0,1)) is totally bounded ( image of the open ball, radius 1, centered at 0)

2)cl(T(B(0,1)) is compact

3,For every bounded sequence $x_n$, $Tx_n$ has a convergent subsequence

I think you can use 1) for the above problem.

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