Spectral Properties of Laplacians on an Abstract Wiener Space with a Weighted Wiener Measure

We study the Hodge–Kodaira Laplacians on an abstract Wiener space with a weighted Wiener measure. Under some exponential integrability conditions for the density function, we determine the kernels, establish the relation between spectrum sets and essential spectrum sets, and prove the Hodge–Kodaira decomposition theorem.