Is this operator on $L^2$ compact?

Question

Let $T: L^2([0,1]) rightarrow L^2([0,1])$ be a linear and bounded
operator defined as: $$Tu(t) = log(1+t)u(t)
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 tin[0,1]$$
Is $T$ compact? Justify the answer.

Hi, I'm having a lot of issues trying to understand if an operator in $L^2$ is compact or not. May u help me by telling me what is the optimal procedure in order to understand if a given operator in $L^2$ is compact? Many thanks!

Martin Argerami · Answer

There is no "optimal procedure", as there are lots of ways to express an operator.
One common way to tell if an operator is compact, is to show that it is a norm limit of finite-rank operators.
On the opposite direction, the spectrum of a compact operator is always countable.
For a multiplication operator like yours, it is an easy-to-establish fact that the spectrum of an operator of the form $ulongmapsto f,u$ is the closure of the image of $f$. That means that, over $[0,1]$, whenever your operator is multiplication by a continuous function its spectrum will be uncountable.

Is this operator on $L^2$ compact?

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