Maximal sectorial operators and invariant operator ranges

Abstract

For unbounded maximal sectorial operators we establish necessary and sufficient conditions for the domain equality domA=domA⁎ and for the equality ReA=AR of operator real part ReA and form real part AR. Here ReA=12(A+A⁎) is half of the operator sum defined on domA∩domA⁎, whereas AR=12(A+˙A⁎) is the self-adjoint operator given by half of the form-sum of A and A⁎ so that, in general, ReA⊆AR. The natural question posed in [6], whether for a maximal sectorial operator A the equalitydomA=domA⁎implies the equalityReA=AR, is answered negatively in this paper. We construct families of unbounded coercive m-sectorial operators A such that domA=domA⁎ for which ReA is a closed symmetric non-selfadjoint operator or a non-closed essentially selfadjoint operator. Moreover, we show that the domain equalities domA=domA⁎ and domReA=domAR are equivalent to problems of invariant operator ranges of bounded selfadjoint or unitary operators as well as to the existence of bounded operators with specific operator range properties.