On operators whose adjoints or second adjoints attain their norms

Abstract

A long-standing open problem asks whether there exists an infinite-dimensional Banach space on which every bounded linear operator attains its norm. With $\mathrm{NA}_1(X,Y)$ and $\mathrm{NA}_2(X,Y)$ denoting the classes of operators whose adjoints and second adjoints, respectively, attain their norms, we prove that \[ \mathrm{NA}_2(c_0,c_0)=\mathcal{L}(c_0,c_0) \qquad\text{and}\qquad \mathrm{NA}_2(\ell_1,\ell_1)=\mathcal{L}(\ell_1,\ell_1), \] providing, to the best of our knowledge, the first known infinite-dimensional spaces on which every operator has a norm-attaining second adjoint. Building on the result for $c_0$, we undertake a systematic study of this equality within a natural family of $\ell_1$-preduals given by hyperplanes of $c$, obtaining a complete characterization in this setting. In particular, we prove that \[ \mathrm{NA}_2(c,c)\neq \mathcal{L}(c,c), \qquad\text{whereas}\qquad \mathrm{NA}_3(c,c)=\mathcal{L}(c,c). \] We also establish Holub--Mujica-type theorems for the classes $\mathrm{NA}_1$ and $\mathrm{NA}_2$. More precisely, under suitable separability and approximation property assumptions, the identity $\mathcal L(X,Y)=\mathrm{NA}_1(X,Y)$ forces every operator from $X$ into $Y$ to be compact, whereas $\mathcal L(X,Y)=\mathrm{NA}_2(X,Y)$ forces every weakly compact operator from $X$ into $Y$ to be compact. Finally, strengthening a construction of Ostrovskii, we show that every infinite-dimensional Banach space admits an equivalent norm and a projection whose second adjoint does not attain its norm.