First Order Selfadjoint Differential Operators with Involution

In this paper, certain spectral properties related with the first order linear differential-operator expression with involution in the Hilbert space of vector-functions at finite interval have been examined. Firstly, the minimal and maximal operators which are generated by the first order linear differential-operator expression with involution in the Hilbert spaces of vector-functions has been described. Then, the deficiency indices of the minimal operator have been calculated. Moreover, the space of boundary values of the minimal operator have been constructed. Afterwards, by using the method of Calkin–Gorbachuk, the general form of all selfadjoint 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.

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