Let's consider a compact operator $T: X \rightarrow l^{p}, 1 \leq p < \infty$. I would like to check, whether it's possible to approximate $T$ by the operators of a finite rank with respect to an operator norm.
For Hilbert spaces it's known that any compact operator can be approximated by the finite rank operator with respect to operator norm (moreover, the statement holds for any space that admits Schauder basis). For arbitrary Banach spaces the property does not hold (there are some counterexamples constructed).
How to check, if in that particular case the approximation property holds?
Any help would be much appreciated.
A Banach space $E$ has the approximation property if and only if for every Banach space $X$ every compact operator $T\colon X\to E$ is approximable by finite-rank operators. Certainly, $\ell_p$ has the approximation property having a Schauder basis.
For the proof of the general statement see, e.g., Proposition 4.12 in Ryan's Introduction to Tensor Products of Banach Spaces.