On spectral synthesis for dissipative Dirac type operators (Q405701)

In this paper, the authors consider the first order system \[ -iB^{-1}y'+Q(x)y=\lambda y,\;x\in [0,1],\tag{1} \] where \(B\) and \(Q\) are \(n\times n\) matrices such that \(B\) is a nonsingular diagonal matrix with complex entries and \(Q(x)\in L^1([0,1];\mathbb C^{n\times n})\), \(y=(y_1,\dots,y_{n})^{t}\), subject to the conditions \[ Cy(0)+Dy(1)=0,\tag{2} \] where \(C,D\in \mathbb C^{n\times n}\). For \(B=B^{*}\) with \(\text{rank}(C,D)=n\), the problem (1), (2) generates a dissipative operator \(L_{C,D}(Q)\). Using the resolvent operator of the dissipative operator \(L_{C,D}(Q)\), the authors obtain that, if the dissipative operator is a complete maximal dissipative (accumulative) operator, then, for any \(\lambda\) belonging to the lower (upper) half plane, the inverse operator \((L_{C,D}(Q)-\lambda)^{-1}\) exists and admits spectral synthesis.