Place : Math Bldg 213

Title: Norm attainment on the Birkhoff-James orthogonality 

Abstrct: In this talk, we study the Birkhoff-James orthogonality and the related Bhatia-\v{S}emrl property (in short, B\v{S}p) to discuss the quantity of operators with such property in the whole operator space, $\mathcal{L}(X,Y)$. We show that when $X$ has either property quasi-$\alpha$ or the RNP, the set of operators with B\v{S}p is dense in $\mathcal{L}(X,Y)$. Moreover, we focus on the case when $X=c_0$ to see that the operators with B\v{S}p defined on $c_0$ do not play well. More precisely, we prove first in many cases that there is no nonzero operator with B\v{S}p, and secondly that for every Banach space $Y$ the set of operators with B\v{S}p in $\mathcal{L}(X,Y)$ is never dense.