Reduction of generalized resolvents of linear operator functions (Q1881583)

In this paper, linear operator functions, that is, functions of the form \(L(\lambda)=T-\lambda S\), \(\lambda\in \mathbb C\), where \(S\) and \(T\) are linear bounded operators between Banach spaces, are studied. A generalized (or relative) inverse of \(L\), denoted by \(L^+\), is called a generalized resolvent of \(L\) if it is smooth at \(0\) and the spaces \(R(L^+(\lambda))\) and \(N(L^+(\lambda))\) do not depend on \(\lambda\). The main result of this paper is a decomposition theorem for the generalized resolvent. The reducing projections and the involved projection spaces are analysed. Finally, the results are applied to two special situations: the commutative case and the case of unbounded operators.