The Construction of Regular Spaces and Hyperspaces with Respect to a Particular Operator

Abstract

Let H be a Hilbert space with an inner product (·,·) and corresponding norm ║·║. There is given an unbounded operator B : D(B) ⊂ H → H such that 0 ∈ (B) and – B is a generator of an analytic semigroup. That is, $$\text{p}(\text{B}) \supset \sum {^ + = \{ \lambda |\lambda \in {\not {\text C}},\,0 < \omega < |\arg \lambda |\mathop < \limits_ = \pi \} \cup \{ 0\} }$$ and $$\left\| {{\text{R}}\left( {\lambda ;{\text{B}}} \right)} \right\| {\mathop < \limits_ = } \frac{{\text{M}}} {\lambda },\forall \lambda \in \sum ^ +$$ where ω < π/2 and M is a positive constant, see, e.g., [4]. Note that, in this case, B* also satisfies conditions (1.1) and (1.2), hence – B* also generates an analytic semigroup e–tB* = (e–tB)*. For such an operator B, powers of arbitrary order can be defined an they enjoy nice properties. We shall collect some of them as follows (Conf., e.g., [4])