Q-functions and boundary triplets of nonnegative operators

Operator-valued $Q$-functions for special pairs of nonnegative selfadjoint extensions of nonnegative not necessarily densely defined operators are defined and their analytical properties are studied. It is shown that the Kre\u\i n-Ovcharenko statement announced in \cite{KrO2} is valid only for $Q$-functions of densely defined symmetric operators with finite deficiency indices. A general class of boundary triplets for a densely defined nonnegative operator is constructed such that the corresponding Weyl functions are of Kre\u\i n-Ovcharenko type

Everything is possible for the domain intersection dom T \cap dom T*

This paper shows that for the domain intersection \dom T\cap\dom T^* 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 \dom T\cap\dom T^*=\{0\}, we construct classes of operators for which \dim(\dom T\cap\dom T^*)= n \in \dN_0; \dim(\dom T\cap\dom T^*)= \infty and at the same time \codim(\dom T\cap\dom T^*)=\infty; and \codim(\dom T\cap\dom T^*)= n \in \dN_0; the latter includes~the case that \dom T\cap\dom T^* is dense but no core of $T$ and $T^*$ and the case \dom T=\dom T^* 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)=\dC, $T$ 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.Comment: 34 page