Birkhoff-James orthogonality and its local symmetry in some sequence spaces (Q6042188)

Let \(\mathbb{X}\) be a Banach space. Given any two elements \(x, y \in \mathbb{X},\) \(x\) is said to be Birkhoff-James orthogonal to \(y\) if \(\|x + \lambda y\| \geq \|x\|,\) for any scalar \(\lambda\); this is denoted as \(x \perp_B y.\) Birkhoff-James orthogonality, in general, is not symmetric, i.e., \(x \perp_B y\) does not imply \(y \perp_B x.\) An element \(x\) is called left symmetric point if \(x \perp_B y\) implies that \(y \perp_B x\) for all \(y \in \mathbb{X}.\) Similarly, a point \( x \in \mathbb{X}\) is said to be a right-symmetric point if \(y \perp_B x\) implies that \(x \perp_B y\) for all \(y \in \mathbb{X}.\) In this article, Birkhoff-James orthogonality is studied for sequence spaces. The authors present complete and tractable characterizations for any two elements \(x=(x_n)_{n \in \mathbb{N}}, y =(y_n)_{n \in \mathbb{N}} \) of \(\ell_{\infty}, c, c_0, c_{00}, \ell_1, \ell_p\) spaces separately. Using these characterizations, the authors characterize the smooth points of the respective sequence spaces. The authors also show that, for the sequence spaces \(\ell_{\infty}, c, c_0, c_{00},\) the only nonzero left symmetric points are scalar multiples of \(e_n,\) where \(e_n\) denotes the sequence having the \(n\)-th term \(1\) and the rest of the terms \(0.\) It is shown that, for the spaces \(\ell_{\infty}\) and \(c,\) \(x=(x_n)_{n \in \mathbb{N}}\) is a right symmetric point if and only if \(|x_n|=\|x\|,\) for any \(n \in \mathbb{N}.\) Moreover, \(c_0\) and \(c_{00}\) have no nonzero right symmetric points. For the space \(\ell_1,\) it is shown that there are no nonzero left symmetric points, whereas the only nonzero right symmetric points are the scalar multiples of \(e_n,\) for each \(n \in \mathbb{N}.\) Using the fact that left symmetric and right symmetric points are invariant under an onto isometry, it is shown that \(T\) is an onto linear isometry on \(\ell_p,\) where \(1 \leq p \neq 2 \leq \infty\) if and only if \(T\) is a signed permutation operator.