Norm attaining operators and variational principle
Volume 265 / 2022
Studia Mathematica 265 (2022), 343-360 MSC: Primary 46B20, 47L05, 28A05; Secondary 47B48, 26A21. DOI: 10.4064/sm210628-6-9 Published online: 4 March 2022
We establish a linear variational principle extending Deville–Godefroy–Zizler’s one. We use this variational principle to prove that if $X$ is a Banach space having property $(\alpha )$ of Schachermayer and $Y$ is any Banach space, then the set of all strongly norm attaining linear operators from $X$ into $Y$ is the complement of a $\sigma $-porous set. Moreover, we apply our results to an abstract class of (linear and nonlinear) operator spaces.