Nonlinear equivalence of Banach spaces based on Birkhoff-James orthogonality
Nonlinear equivalence of Banach spaces based on Birkhoff-James orthogonality
复制标题
DOI：
10.1016/j.jmaa.2021.125444
复制
发表时间：
--
影响因子：
1.3
通讯作者：
Ryotaro Tanaka
中科院分区：
文献类型：
--
作者：
Ryotaro Tanaka
如果 X 和 Y 之间存在（可能是非线性的）双射，并且在两个方向上都保持 Birkhoff-James 正交性，则称 Banach 空间 X 与另一个 Y 就 Birkhoff-James 正交性结构同构（用 X∼ B J Y 表示）。结果表明，如果 X 或 Y 是有限维且 X∼ B J Y，则 X≅ Y，并且如果 1< p< q<∞，则 ℓ p≁ B J ℓ q。此外，如果H是一个Hilbert空间，且dim H≥ 3且H∼ B J X，则H= X。在二维情况下，证明ℓ p, q 2∼ B J ℓ 2 2，这表明Banach空间之间的非线性Birkhoff-James正交性保持子不一定是等距同构的标量倍数。
A Banach space X is said to be isomorphic to another Y with respect to the structure of Birkhoff-James orthogonality, denoted by X∼ B J Y, if there exists a (possibly nonlinear) bijection between X and Y that preserves Birkhoff-James orthogonality in both directions. It is shown that X≅ Y if either X or Y is finite dimensional and X∼ B J Y, and that ℓ p≁ B J ℓ q if 1< p< q<∞. Moreover, if H is a Hilbert space with dim H≥ 3 and H∼ B J X, then H= X. In the two-dimensional case, it turns out that ℓ p, q 2∼ B J ℓ 2 2, which indicates that nonlinear Birkhoff-James orthogonality preservers between Banach spaces are not necessarily scalar multiples of isometric isomorphisms.