Let \(\Gamma ,\Delta \) be nonempty index sets, and let H, K be inner product spaces. We prove that for \(p\ge 1\) any surjective phase-isometry between \(\ell ^p(\Gamma ,H)\) and \(\ell ^p(\Delta , K)\) is a plus–minus linear isometry. This can be considered as an extension of Wigner’s theorem for real \(\ell ^p(\Gamma , H)\)-type spaces.

中文翻译：

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两个$$ \ ell ^ p（\ Gamma，H）$$ℓp（Γ，H）型空间之间的相位等距

令\（\ Gamma，\ Delta \）为非空索引集，令H，  K为内积空间。我们证明，对于\（p \ ge 1 \），在\（\ ell ^ p（\ Gamma，H）\）和\（\ ell ^ p（\ Delta，K）\）之间的任何射影相位等距都是一个加号–减去线性等距。这可以看作是Wigner定理对实\（\ ell ^ p（\ Gamma，H）\）型空间的扩展。