Everything is possible for the domain intersection dom T ∩ dom T⁎

Abstract

In this paper we show that for the domain intersection domT∩domT⁎ of a closed linear operator and its Hilbert space adjoint everything is possible for very common classes of operators with non-empty resolvent set. Apart from the most striking case of a maximal sectorial operator with domT∩domT⁎={0}, we construct classes of operators for which dim(domT∩domT⁎)=n∈N0; dim(domT∩domT⁎)=∞ and at the same time codim(domT∩domT⁎)=∞; and codim(domT∩domT⁎)=n∈N0; the latter includes the case that domT∩domT⁎ is dense but no core of T and T⁎ and the case domT=domT⁎ for non-normal T. We also show that all these possibilities may occur for operators T with non-empty resolvent set such that either W(T)=C, T is maximal accretive but not sectorial, or T is even maximal sectorial. Moreover, in all but one subcase T can be chosen with compact resolvent.