4 problems

The conjecture. Birkhoff-James orthogonality is C-approximately symmetric in $X$ if and only if the local property (P) holds for each $x\in\operatorname{Ext} B_X$.

Let $\lVert\cdot\rVert$ be any norm on $\mathbb{C}^n$ or $\mathbb{R}^n$, and let $\lVert\cdot\rVert'$ be the corresponding induced norm on $M_n(\mathbb{C})$ or $M_n(\mathbb{R})$, r…

Bhatia–Šemrl conjecture. If $A\perp_B B$ in $\|\cdot\|_{\nu}$, then there exists a unit vector $y\in\mathcal{M}_{n\times 1}$ with $\nu(y)=1$ and $\nu(Ay)=\|A\|_{\nu}$ such that

Let $\mathbb{X}$ be a finite-dimensional real Banach space. For $x,y\in\mathbb{X}$, write $x\perp_B y$ when $\|x+\lambda y\|\geq\|x\|$ for every real $\lambda$. An operator…