Abstract extensions of functional calculi
We describe a ‘minimal’ and a ‘maximal’ method of extending a certain basic functional calculus towards unbounded operators. The maximal method is a generalisation of (and consistent with) the approach via regularisation, favoured in the author’s earlier papers, and it covers also Balakrishnan’s calculus for generators of $C_0$-semigroups. It is shown that for many special cases the maximal and the minimal extensions coincide. However, some simple but rather startling examples are devised to show not only that the two extensions may differ, but also that each of them may yield an intuitively ’wrong’ result.