This paper studies multi-operator wireless communication systems aided by general reconfigurable intelligent surface (RIS), including both conventional single-connected RIS and beyond-diagonal RIS (BD-RIS). Specifically, we consider a system where multiple operators coexist in the same area over different frequency bands, each with a single-antenna base station, while one operator serves its single-antenna user with the aid of an RIS. In such a system, the RIS may unintentionally reflect signals from the non-serving operators, leading to inter-operator interference and rapid fluctuations of their effective channels. To address this issue, we propose a practical RIS design framework that maximizes the received signal power of the serving operator while enforcing fixed RIS-reflected channels of the non-serving operators. We derive closed-form solutions to the resulting optimization problem, based on a novel technique to deal with the coupled unitary and linear equality constraints. We further give scaling law analysis of the received signal power. For a two-operator system, the received signal power scales quadratically with the number of RIS elements for group-connected BD-RIS with group size Gs>=2, whereas for conventional single-connected RIS it scales only linearly. More generally, for an L-operator system with L-1 non-serving operators, the scaling-law transition occurs at Gs=L, where quadratic scaling is achieved when Gs>=L, and linear scaling otherwise. These results demonstrate that, in a multi-operator system, quadratic scaling is achievable only with BD-RIS architectures having enough interconnections. Simulation results validate the analysis and show the significant gain of BD-RIS over conventional RIS in multi-operator systems. In particular, group-connected BD-RIS with Gs=2 achieves a 13dB gain over conventional RIS in a two-operator system with a 128-element RIS.