Let $T\colon \mathbb L^p[0,1]\to \mathbb L^p[0,1]$, $1<p<+\infty$, be a linear bounded operator such that $\operatorname{Im}(T)$ is contained in the space of continuous functions. It was shown in this thread that $T$ is a compact operator.

Some non trivial example of such $T$ can be given, for example kernel operators with appropriated conditions on the kernel. The question is whether there exists other example (maybe asking for a characterisation is too ambitious).

  • $\begingroup$ From your comment in the linked post, doesn't it suffice to consider operators from $L_p[0,1]$ to $C[0,1]$? This may help: Operators from $X$ to $C[0,1]$ are characterized by functions from $[0,1]$ to $X^*$ that are continuous when $X^*$ has the weak* topology (c.f. Dunford and Schwartz, Linear Operators, vol 1, Theorem VI.7.1). $\endgroup$ – David Mitra May 17 at 13:03
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    $\begingroup$ Operator $f\mapsto \int_0^x f(t)dt$ $\endgroup$ – Boris Bilich May 17 at 13:17
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    $\begingroup$ Another examples: Fractional integrals of the form $(- \partial_{tt}^2 )^{-\beta}$ for any $\beta > 1/p$, by Morrey's Inequality. $\endgroup$ – Adrián González-Pérez May 18 at 13:34

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