Customized static operator design has enabled widespread application of Evolutionary Algorithms (EAs), but their search effectiveness often deteriorates as evolutionary progresses. Dynamic operator configuration approaches attempt to alleviate this issue, but they typically rely on predefined operator structures and localized parameter control, lacking sustained adaptive optimization throughout evolution. To overcome these limitations, this work leverages Large Language Models (LLMs) to perceive evolutionary dynamics and enable operator-level meta-evolution. The proposed framework, LLMs for online operator design in Evolutionary Optimization, named LLM4EO, comprises three components: knowledge-transfer-based operator design, evolution perception and analysis, and adaptive operator evolution. Firstly, operators are initialized by leveraging LLMs to distill and transfer knowledge from well-established operators. Then, search behaviors and potential limitations of operators are analyzed by integrating fitness performance with evolutionary features, accompanied by suggestions for improvement. Upon stagnation of population evolution, an LLM-driven meta-operator dynamically optimizes gene selection of operators by prompt-guided improvement strategies. This approach achieves co-evolution of solutions and operators within a unified optimization framework, introducing a novel paradigm for enhancing the efficiency and adaptability of EAs. Finally, extensive experiments on multiple benchmarks of flexible job shop scheduling problem demonstrate that LLM4EO accelerates population evolution and outperforms tailored EAs.