A three-point normal boundary value problem for an operator-differential equation (Q1920820)

A densely defined closed operator \(B\) in a Hilbert space is called formally normal if \(D(B)\subseteq D(B^*)\) and \(|Bf|=|B^*f|\) for all \(f\in D(B)\). A formally normal operator is called maximal formally normal if it has no trivial formally normal extensions. The aim of this paper is to describe maximal formally normal extensions of the closure in \(L_2((0,1);H)\) of an operator of the form \(L_0'u=u'(t)+A(t)u(t)\), where \(A(t)=A_1\) for \(0\leq t<h\), \(A(t)=A_2\) for \(h\leq t\leq 1\), \(A_1\) and \(A_2\) are selfadjoint on \(H\), \(D(A_1)=D(A_2)=D(A)\), \(A_2\geq 0\), \(A_1\geq A_2+E\) and \(0<h\leq 1\). The description is done in terms of three-point boundary conditions.