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.

  • $\begingroup$ Does $l^p$ admit a Schauder basis? $\endgroup$ – Daniel Fischer Dec 15 '15 at 22:37
  • $\begingroup$ @DanielFischer Sure, it admits monotone Schauder basis, consisting of unit vectors. Moreover, any orthonormal basis is a Schauder basis in $l^{p}$, since it's separable. $\endgroup$ – hyperkahler Dec 15 '15 at 22:45
  • $\begingroup$ Oh, I think I misunderstood the question. You want a strategy to (explicitly) find finite-rank approximations for a given compact operator? $\endgroup$ – Daniel Fischer Dec 15 '15 at 22:48
  • $\begingroup$ @DanielFischer I'm familiar with the result, stating that the space of compact operators on a Hilbert space $K(H)$ coincides with the closure of a space of all bounded finite-dimensional operators. In case $H$ is separable, for a taken basis $\{e_{n} \}$ we can consider $P_{n} $ - a projector onto $span\{e_{1}, \ldots, e_{n} \}$. Then, for any $K \in K(H)$, we'll get $||K-P_{n}K|| \to 0$. Seems that here we're considering operators from $H$ to itself, whereas in the situation described in the question is different. $\endgroup$ – hyperkahler Dec 15 '15 at 22:57
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    $\begingroup$ Ah, your problem is that here the domain of the operator is arbitrary? Doesn't matter a bit, same construction. $\endgroup$ – Daniel Fischer Dec 15 '15 at 23:05

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.


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