Title: A numerical range approach to Birkhoff-James orthogonality with applications Show Chinese title

Title: 数值域方法在Birkhoff-James正交性中的应用

Abstract: The main aim of this paper is to provide characterizations of Birkhoff-James orthogonality (BJ-orthogonality in short) in a number of families of Banach spaces in terms of the elements of significant subsets of the unit ball of their dual spaces, which makes the characterizations more applicable. The tool to do so is a fine study of the abstract numerical range and its relation with the BJ-orthogonality. Among other results, we provide a characterization of BJ-orthogonality for spaces of vector-valued bounded functions in terms of the domain set and the dual of the target space, which is applied to get results for spaces of vector-valued continuous functions, uniform algebras, Lipschitz maps, injective tensor products, bounded linear operators with respect to the operator norm and to the numerical radius, multilinear maps, and polynomials. Next, we study possible extensions of the well-known Bhatia-\v{S}emrl theorem on BJ-orthogonality of matrices, showing results in spaces of vector-valued continuous functions, compact linear operators on reflexive spaces, and finite Blaschke products. Finally, we find applications of our results to the study of spear vectors and spear operators. We show that no smooth point of a Banach space can be BJ-orthogonal to a spear vector of $Z$. As a consequence, if $X$ is a Banach space containing strongly exposed points and $Y$ is a smooth Banach space with dimension at least two, then there are no spear operators from $X$ to $Y$. Particularizing this result to the identity operator, we show that a smooth Banach space containing strongly exposed points has numerical index strictly smaller than one. These latter results partially solve some open problems. Show Chinese abstract

Abstract: 本文的主要目的是通过其对偶空间单位球中显著子集的元素来刻画若干族Banach空间中的Birkhoff-James正交性（简称为BJ-正交性），这使得这些刻画更具应用价值。 为此所用的工具是对抽象数值域及其与BJ-正交性的关系进行精细研究。 在其他结果中，我们提供了向量值有界函数空间的BJ-正交性的一个刻画，该刻画基于定义域和目标空间的对偶空间，此结果被应用于向量值连续函数空间、一致代数、Lipschitz映射、注入张量积、关于算子范数和数值半径的有界线性算子、多线性映射以及多项式的研究。 接下来，我们研究了关于矩阵的BJ-正交性经典Bhatia-Šemrl定理的可能推广，得到了向量值连续函数空间、自反空间上的紧线性算子以及有限Blaschke乘积空间中的相关结果。 最后，我们将我们的结果应用于spear向量和spear算子的研究。 我们证明，Banach空间中的光滑点不可能与空间$Z$中的spear向量BJ-正交。 由此得出，如果$X$是包含强暴露点的Banach空间，而$Y$是至少二维的光滑Banach空间，则从$X$到$Y$不存在spear算子。 将这一结果特化到恒等算子上，我们证明了包含强暴露点的光滑Banach空间的数值指标严格小于1。 后一结果部分解决了某些未解决的问题。