  The notion of quasi boundary triples and their Weyl functions from extension
theory of symmetric operators is extended to the general framework of adjoint
pairs of operators under minimal conditions on the boundary maps. With the help
of the corresponding abstract Titchmarsh-Weyl $M$-functions sufficient
conditions for the unique solvability of the related boundary value problems
are obtained and the solutions are expressed via Krein-type resolvent formulae.
The abstract theory developed in this manuscript can be applied to a large
class of elliptic differential operators.
