关于循环性猜想

中文摘要 AI 辅助

我们证明了复Banach格上每个正算子的外围谱是循环的，从而在没有额外增长假设的情况下解决了循环性猜想。我们的证明基于一个由Vesentini关于谱半径次调和性定理推导出的谱支配定理：当两个正算子具有相同的谱半径且其中一个支配另一个时，较小算子的外围谱包含在较大算子的外围谱中。这推广了Räbiger和Wolff的一个早期结果。在过渡到合适的格扩张后，基于挠算子（torsion operators）的平均化过程产生一个递减的正小算子序列，该序列保持谱半径并渐近地旋转自相似。一个超幂（ultrapower）论证将此渐近关系转化为精确的旋转自相似性，使我们能够从谱支配推导出循环性。同样的方法也为Lotz关于Abel可解算子的循环性定理提供了另一种证明。

英文摘要

We prove that the peripheral spectrum of every positive operator on a complex Banach lattice is cyclic, thereby resolving the cyclicity conjecture without additional growth assumptions. Our proof is based on a spectral domination theorem derived from Vesentini's theorem on the subharmonicity of the spectral radius: when two positive operators have the same spectral radius and one dominates the other, the peripheral spectrum of the smaller operator is contained in that of the larger one. This extends an earlier result of Räbiger and Wolff. After passing to a suitable lattice extension, an averaging procedure based on torsion operators produces a decreasing sequence of positive minorants which retain the spectral radius and become asymptotically rotationally self-similar. An ultrapower argument turns this asymptotic relation into exact rotational self-similarity, allowing us to deduce cyclicity from spectral domination. The same approach also yields an alternative proof of Lotz's cyclicity theorem for Abel solvable operators.

发表机构

• Christian-Albrechts-Universität zu Kiel（基尔大学（克里斯蒂安-阿尔布雷希茨大学））

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