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For a countable subset $Z$ of a metric space $X$, consider the Banach space $\ell^p(Z,\ell^p)$ for $p\in[1,\infty)$. The space of continuous functions $C_0(X)$ acts on $\ell^p(Z,\ell^p)$ by multiplication (restricted to $Z$). Let $T$ be a compact operator on $\ell^p(Z,\ell^p)$, regarded as a $Z$-by-$Z$ matrix of bounded operators on $\ell^p$. Is it necessarily the case that for any bounded subset $B\subset X$, the set of $(x,y)\in B\times B$ such that $T_{xy}\neq 0$ is finite?

(The background behind this question is that I am comparing two definitions:

  1. $fT$ and $Tf$ are compact operators on $\ell^p(Z,\ell^p)$ for all $f\in C_0(X)$.

  2. Each $T_{xy}$ is a compact operator on $\ell^p$, and for each bounded subset $B\subseteq X$, the set of $(x,y)\in B\times B$ such that $T_{xy}\neq 0$ is finite.)

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