Spectrum of differential operators with elliptic adjoint on a scale of localized Sobolev spaces

In this paper, we provide a complete study of the spectrum constant coefficients differential operator on scale localized Sobolev spaces, $$H^{s}_{\mathrm{loc}}(I),$$ which are Fréchet spaces. This is quite different from what find in literature, where all relevant results concerned with Banach Our aim to understand behavior three types (point, residual, and continuous) relation between them those dual operator. The main result present shows that there no complex number resolvent set such operators, suggest new way define if want reproduce classical theorems Spectral Theory