An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs

We consider infinite dimensional Kolmogorov equations in a separable Hilbert space $H$ having singular first order terms. We prove an optimal regularity result for solutions to such equations. This result allows to study semilinear SPDEs of the form $ dX_t = A X_t dt + (-A)^{gamma}F(X_t)dt + dW_t $ driven by a cylindrical Wiener process $W = (W_t)$; here $A$ is a suitable self-adjoint operator on $H$