Functional inequalities for uniformly integrable semigroups and application to essential spectrums

Fu-Zhou Gong, Feng-Yu Wang
2002 Forum mathematicum  
Let (E, F, µ) be a probability space, (E, D(E)) a (not necessarily symmetric) Dirichlet form on L 2 (µ), and P t the associated sub-Markov semigroup. The equivalence of the following eight properties is studied: (i) the L 2 -uniform integrability of the unit ball in the Sobolev space; (ii) the super-Poincaré inequality (1.2); (iii) the F -Sobolev inequality (1.3); (iv) the L 2 -uniform integrability of P t ; (v) the L 2uniform integrability of the associated resolvents; (vi) the compactness of
more » ... the compactness of P t ; (vii) the compactness of the associated resolvents; (viii) empty essential spectrum of the associated generator. The main results can be summarized as follows. In general, (i), (ii) and (iii) are equivalent to each other, and they imply (iv) which is equivalent to (v). If P t has transition density and F is µ-separable, then the first seven properties from (i) to (vii) are equivalent. If in addition (E, D(E)) is symmetric, then all the above eight properties are equivalent. Moreover, the essential spectrum of the generator is estimated by using weaker functional inequalities.
doi:10.1515/form.2002.013 fatcat:4qzf6h63vngdliujlpqsb4qzlu