酉群中对称模的短路
Short paths for symmetric norms in the unitary group
摘要：
对无限维希尔伯特空间H上给定的赋对称范理想I，研究了经典Banach-李酉群$$ U_I={u is a unitary operator in H, u-1\in I}. $$中的可求长距离.证明了：只要U_I的单参数子群的指数谱由$π$有界，且U_I中的任意两个元素可以用一条短路连接，从而获得一个霍普夫-里诺定理在这个无限维的设置，为广泛的和相关的类（不一定光滑）度量。证明了在对称范数为严格凸的条件下，单参数群是连接给定端点的唯一短路。
英文摘要：
For a given symmetrically normed ideal I on an infinite dimensional Hilbert space H, we study the rectifiable distance in the classical Banach-Lie unitary group $$ U_I={u is a unitary operator in H, u-1\in I}. $$ We prove that one-parameter subgroups of U_I are short paths, provided the spectrum of the exponent is bounded by $π$, and that any two elements of U_I can be joined with a short path, thus obtaining a Hopf-Rinow theorem in this infinite dimensional setting, for a wide and relevant class of (non necessarily smooth) metrics. Then we prove that the one-parameter groups are the unique short paths joining given endpoints, provided the symmetric norm considered is strictly convex.
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