Preface The influence of the theory of linear evolution equations upon developments in other branches of mathematics, as well as physical sciences, would be hard to exaggerate. The theory has a rich interplay with other subjects in functional analysis, stochastic analysis and mathematical physics. Of par- ticular interest are evolution equations associated with second-order elliptic operators in divergence form. Such equations arise in many models of phys- ical phenomena; the classical heat equation is a prototype example. They are also of interest for nonlinear analysis; the proof of existence of local solutions to many nonlinear partial differential equations uses linear theory. The theory for self-adjoint second-order elliptic operators is well docu- mented, and there is an increasing interest in the non-self-adjoint case. It is one of the aims of the present book to give a systematic study of L p theory of evolution equations associated with non-self-adjoint operators A = - x j a kj x k + k b k k,j + (c k .) + a 0 . x k x k We consider operators with bounded measurable coefficients on arbitrary domains of Euclidean space. The sesquilinear form technique provides the right tool to define such operators, and associates them with analytic semi- groups on L 2 . We are interested in obtaining contractivity properties of these semigroups as well as Gaussian upper bounds on their associated heat ker- nels. Gaussian upper bounds are then used to prove several results in the L p -spectral theory. A special feature of the present book is that several important properties of semigroups are characterized in terms of verifiable inequalities concern- ing their sesquilinear forms. The operators under consideration are subject to various boundary conditions and do not need to be self-adjoint. We also consider second-order elliptic operators with possibly complex-valued co- efficients. Such operators have attracted attention in recent years as their associated heat kernels do not have the same properties as those of their analogues with real-valued coefficients. This book is also motivated by new developments and applications of Gaussian upper bounds to spectral theory. A large number of the results given here have been proved during the last