Maximally Dissipative Differential Operators of First Order in the Weighted Hilbert Space

In this paper, certain spectral properties related with the first order linear differential expression in the weighted Hilbert space at finite interval have been examined. Firstly, the minimal and maximal operators which are generated by the first order linear differential expression in the weighted Hilbert space have been determined. Then, the deficiency indices of the minimal operator have been calculated. Moreover, a space of boundary values of the minimal operator has been constructed. Afterwards, by using the Calkin–Gorbachuk’s method, the general form of all maximally dissipative extensions of the minimal operator in terms of boundary values has been found. Later on, the structure of spectrum of these extensions has been investigated.