The spectral properties of the first order differential operator

The spectral properties of the first order differential operator were studied in L2 Hilbert space.The properties of d/dx differential operator, such as being unbounded, closed or closable, were examined separately and the closure of the operator was determined. The spectrum of the sum of this closure and multiplication by a function operator was studied.It was proved that in L2 space, the differential operator id/dx was self adjoint and unitary equivalent with the multiplication operator. Also its self adjoint extensions were studied in L2 (0,1) space and it was shown that it hasn?t any self adjoint extensions in L2 space.Keywords: Spectrum, differential operator, unitary equivalent.