On the first mixed boundary value problems for linear equations with generalized right invertible operators (Q1862699)

An operator \(V\) belonging to the set of all generalized invertible operators \(W(X)\subset L(X)\), where \(X\) is a linear space over a field of scalar \({\mathcal K}\), is said to be generalized right invertible (GR-invertible) if there is an operator \(W\in{\mathcal W}_V\) such that \(\text{Im}(VW-I) \in\text{Ker} V\), i.e., \(VWV=V\), \(V^2W=V\). The author considers the so-called first mixed boundary value problem (FMBVP) with a generalized operator \(V\) for the operator \(Q[V]\) defined as follows: \(Q[V]\): \(\sum^M_{m=0} \sum^N_{n=0} V^mA_{mn} V^nx=y\), \(y \in X\), \(F_jV^jx= y_j\), \(A_{mn}\in L_0(x)\), \(A_{mn}X_{M+N-n} \subset X_m\), \(X_j= \text{dom} V^j\), \(y_j\) are given \((j=0,\dots,M+N-1)\). Following the reviewer and Nguyen Van Mau (for right invertible operators), the author gives necessary and sufficient conditions for solvability of that problem and construction of the resolvent operator. An example is also given.