Linear structures in the set of non-norm-attaining operators on Banach spaces

Abstract

We study large linear structures inside sets arising in the theory of norm-attaining operators. We provide several results in the context of lineability, spaceability, maximal-spaceability, and $(\alpha, \beta)$-spaceability for sets of non-norm-attaining bounded linear operators whenever such sets are nonempty. To be more specific, we show that if $Y$ is a strictly convex renorming of $c_0 (\Gamma)$, then the set $$ \mathcal{L}(c_0 (\Gamma),Y)\setminus \overline{\text{NA} (c_0 (\Gamma),Y)} $$ is $2^{|\Gamma|}$-spaceable. We also prove that $$ \mathcal{L}(d_* (w,1) ,\ell_p )\setminus \overline{\text{NA} (d_* (w,1),\ell_p )} $$ is maximal-spaceable. Finally, we establish that whenever the set of non-norm-attaining operators from a Banach space $X$ into $\ell_p (\Gamma)$ (respectively, $c_0 (\Gamma)$) is nonempty, it contains a subspace linearly isometric to $\ell_p(\Gamma)$ (respectively, $c_0 (\Gamma)$). These results extend and complement several known results in the literature concerning large linear structures in sets of non-norm-attaining operators. Our results are obtained in a more general framework involving group-invariant operators, which allows us to treat classical spaces of operators as special cases.