Summary: Generalized Green’s functions are kernels of integral operators with certain properties. Solution operators with these properties are called Green’s operators. Necessary and sufficient conditions for the existence of Green’s operators are given for the general case, and for operators on Banach spaces (from the author’s summary).
The abstract definition of a Green operator is the following: Let be \((E,E')\) and \((F,F')\) dual pairs of linear spaces and a linear transformation with domain \(D(L)\) in \(E\) and range \(R(L)\) in \(F\), such that \[ \{(y',x') \in F'\times E' \mid \langle Lx,y'\rangle=\langle x,x'\rangle\text{ for all }x\in D(L)\} \tag{1} \] is the graph of a linear operator \(L^0\), \[ \{(x,y) \in X\times Y\mid \langle y,y'\rangle = \langle x,L^0y'\rangle\text{ for all }y'\in D(L^0)\} \tag{2} \] is the graph of \(L\). \(L\) is called a Green operator, if there exists a linear transformation \(T\) with \(D(T) =F\), \(R(T)\subseteq E\), satisfying condition (1), \(D(T^0) = E'\), \(R(T)\subseteq D(L)\) and \(R(T^0)\subseteq D(L^0)\) such that \(TLT = T\) and \(LTL = L\(L\) are satisfied.