Summary:
This Master Thesis essentially consists by two parts. In the first four
chapters we develop the general theory of strongly continuous semigroups of
bounded linear operators, with emphasis on the representation and generation
theorems, particularly those due to Hille-Yosida, Feller-Miyadera-Phillips,
Lumer-Phillips and Stone. Several classes of semigroups are examined, including
the very important for our purpose Gaussian semigroup.
In the remaining three chapters we investigate the problem of spectral
independence of operators acting simultaneously on different Lp-spaces and
arising as generators of strongly continuous semigroups. We use tools such as
compactness, Gaussian estimates and commutator estimates presented in books and
articles by E.B. Davies, W. Arendt, P.C. Kunstmann, M. Hieber and E. Schroche.
The main result can be stated as follows: Given as open subset Ω of IRn and a
family Tp of consistent semigroups on Lp(Ω), independence from p[1,+) of the
spectrum of the generator Ap can be obtained, provided the semigroup T2 is
strongly continuous on L2(Ω) and the integral kernel of the generator A2
satisfies a certain Gaussian estimate of order m. We illustrate the general
theory by presenting concrete applications to the theory of elliptic
differential operators.
Keywords:
Strongly continuous semigroup, Infinitesimal generator, Elliptic differential operator, Lp-spectral independence, Gaussian estimate
