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Given an infinite set $\Gamma$ and an almost disjoint family $\mathcal{A}$ on$\Gamma$, let $Y_{\mathcal{A}}$ denote the closed subspace of$\ell_\infty(\Gamma)$ spanned by the indicator functions $1_{\bigcap_{j=1}^nA_j}$ for $n\in\mathbb{N}$ and $A_1,\ldots,A_n\in\mathcal{A}$. We show that if$\mathcal{A}$ has cardinality greater than $\Gamma$, then $Y_{\mathcal{A}}$contains closed subspaces which cannot be realised as the kernels of anybounded operators $Y_{\mathcal{A}}\rightarrow \ell_{\infty}(\Gamma)$.Consequently the spaces $\ell_{\infty}(\Gamma)$, for any infinite set $\Gamma$,and $C_0(K_{\mathcal{A}})$, where $\mathcal{A}$ is an uncountable almostdisjoint family on $\mathbb{N}$ and $K_{\mathcal{A}}$ the locally compactMr\'{o}wka space associated with $\mathcal{A}$, contain closed subspaces whichdo not arise as the kernels of any bounded operators on them.

Kernels of operators on certain Banach spaces associated with almost disjoint families