We introduce and study modular Birkhoff–James orthogonality for typical Banach modules $$B({\mathbb {X}},{\mathbb {Y}})$$ and $$K({\mathbb {X}},{\mathbb {Y}}),$$ where $${\mathbb {X}}$$ and $${\mathbb {Y}}$$ are Banach spaces. We present some basic characterizations of modular Birkhoff–James orthogonality under certain restrictions, and completely characterize left symmetric points for the said orthogonality relation in $$B({\mathbb {X}},{\mathbb {Y}})$$ and $$K({\mathbb {X}},{\mathbb {Y}}).$$ The complications involved in an analogous study of right symmetric points are illustrated through concrete examples.

中文翻译：

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$$B({\mathbb {X}},{\mathbb {Y}})$$B(X,Y) 和 $$K({\mathbb {X}},{\mathbb {Y}})$$K(X,Y)

我们介绍并研究了典型 Banach 模 $$B({\mathbb {X}},{\mathbb {Y}})$$ 和 $$K({\mathbb {X}},{\ mathbb {Y}}),$$ 其中 $${\mathbb {X}}$$ 和 $${\mathbb {Y}}$$ 是巴拿赫空间。我们给出了一定限制条件下模 Birkhoff-James 正交性的一些基本特征，并在 $$B({\mathbb {X}},{\mathbb {Y}})$$ 和$$K({\mathbb {X}},{\mathbb {Y}}).$$ 通过具体的例子来说明右对称点的类似研究所涉及的复杂性。