Fractional powers of monotone operators in Hilbert spaces
Daniel Hauer, Yuan He, Dehui Liu
Abstract
In this article, we show that if is a maximal monotone operator on with
in the range of , then for every , the Dirichlet problem
associated with the Bessel-type equation
is well-posed for boundary values . This allows us
to define the Dirichlet-to-Neumann (DtN) map associated with as
The existence of the DtN map associated with is the first step
in defining fractional powers of monotone (possibly, nonlinear and multivalued)
operators on . We prove that is monotone on ,
and if is the closure of in
then we provide conditions implying that
generates a strongly continuous semigroup
on . In addition, we show that if is completely accretive,
on for a -finite
measure space , then inherits this
property from .
Keywords:
Monotone operators, Hilbert space, evolution equations, fractional operators.