We consider divergence-form parabolic equation with measurable uniformly elliptic matrix and the vector field in a large class containing, particular, fields $L^p$, $p>d$, as well some that are not even $L_{\loc}^{2+\varepsilon}$, $\varepsilon>0$. establish Hölder continuity of bounded soutions, sharp two-sided Gaussian bound on heat kernel, Harnack inequality.

It is well known that not all operators satisfy Weyl type theorems simultaneously. In this paper, we denote an operator matrix and consider how eight hold for it when certain entry theorems. Moreover, characterize its spectral structure. Finally, the relevant conclusions are promoted to infinite dimensional Hamilton operator.