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On the chain of commuting operators on Banach spaces and Lomonosov's invariant subspace theorem
Tomasz Szczepanski
https://arxiv.org/abs/2507.14297 htt…

On the chain of commuting operators on Banach spaces and Lomonosov's invariant subspace theorem
An operator $T$ on a Banach space is said to be of chain $N$ if there exist non-scalar operators $S_1,...,S_{N-1}$ and a non-zero compact $K$ such that $$T \leftrightarrow S_1 \leftrightarrow S_2 \leftrightarrow ...\leftrightarrow S_{N-1} \leftrightarrow K,$$ where $A\leftrightarrow B$ means $AB=BA$. A connection of this theory to the Lomonosov's Invariant Subspace Theorem is highlighted. It is shown that for every weighted shift $T$ it is of chain $3$. In particular, every non-Lomonosov operat…

Policy-Based Trajectory Clustering in Offline Reinforcement Learning
Hao Hu, Xinqi Wang, Simon Shaolei Du
https://arxiv.org/abs/2506.09202 https://

Policy-Based Trajectory Clustering in Offline Reinforcement Learning
We introduce a novel task of clustering trajectories from offline reinforcement learning (RL) datasets, where each cluster center represents the policy that generated its trajectories. By leveraging the connection between the KL-divergence of offline trajectory distributions and a mixture of policy-induced distributions, we formulate a natural clustering objective. To solve this, we propose Policy-Guided K-means (PG-Kmeans) and Centroid-Attracted Autoencoder (CAAE). PG-Kmeans iteratively trains…

On balanced Hermitian threefolds with parallel Bismut torsion
Quanting Zhao, Fangyang Zheng
https://arxiv.org/abs/2506.15141 https://…

On balanced Hermitian threefolds with parallel Bismut torsion
We continue our study on Hermitian manifolds that are {\em Bismut torsion parallel,} or {\em BTP} for brevity, which means that the Bismut connection has parallel torsion tensor. For $n\geq 3$, BTP metrics can be balanced (and non-Kähler). In this paper, we give a classification of all compact, balanced BTP threefolds.