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 $$\mathcal{L}(d_{\ast }(w,1),{\ell }_p)\setminus \overline{\text{NA}(d_{\ast }(w,1),{\ell }_p)}$$ 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.

30 pages