AbstractOur main goal in this paper is to investigate some spectral properties of all maximally accretive extensions of the minimal operator in the weighted Hilbert spaces of vector functions. We construct the minimal and maximal operators generated by the first order differential operator expression in the weighted Hilbert space of vector functions at finite interval with the use of standard technique. In this case, the minimal operator is accretive but not maximal. Using the Calkin–Gorbachuk method, the general form of all maximally accretive extensions of this minimal operator in terms of boundary conditions is obtained. We also investigate the structure of the spectrum set such maximally accretive extensions of this type of minimal operator.