Some remarks on orthogonality of bounded linear operators-II

Abstract

Let $$\mathbb{X,Y}$$ be normed linear spaces. We continue exploring the validity of the Bhatia–Šemrl (BŠ) Property: “An operator $$T \in \mathbb{L(X,Y)}$$ satisfies Bhatia–Šemrl Property if for any $$A \in \mathbb{L(X,Y)},T \bot_{B} A$$ implies that there exists a unit vector $$x \in \mathbb{X} $$ such that $$\lVert T_{x} \lVert = \lVert T \lVert $$ and $$T_{x}\bot_{B}A_{x}$$ .” A pair of normed linear spaces $$\mathbb{X,Y}$$ is said to be a BŠ pair if for every $$T\in \mathbb{L(X,Y),T}$$ satisfies the BŠ Property if and only if $$M_{T} = D{\cup}(-D)$$ , where D is a non-empty connected subset of $$S_{\mathbb{X}} $$ . We show that $$\ell^{n}_{1},\mathbb{Y}$$ is a BŠ pair for any normed linear space $$\mathbb{Y}$$ , and also obtain some other results in this context. In particular, using the notion of norm attainment set,we characterize the space $$\ell^{3}_{\infty}$$ among all 3-dimensional polyhedral Banach spaces whose unit ball have exactly eight extreme points.