Let \((X,d,\mu )\) be a space of homogeneous type and L be a nonnegative self-adjoint operator on \( L^{2}(X)\) whose heat kernels satisfy Gaussian upper bounds. In this article, we introduce the weighted variable Besov space associated with the operator L and demonstrate that Peetre maximal functions can be used to characterize this space. Furthermore, we provide a detailed study of its atomic decompositions.
