The generator $L$ of the linearized evolution equation of adiabatic oscillations of a gaseous star, ELASO, is a second order integro-differential operator and is realized as a self-adjoint operator in the Hilbert space of square integrable unknown functions with weight, which is the density distribution of the compactly supported background. Eigenvalues and eigenfunctions of the operator $L$ have been investigated from the practical point of view of eigenmode expansion of oscillations. But it should be examined whether continuous spectra are absent in the spectrum of $L$ or not. In order to discuss this question, the existence of essential spectra in a closely related evolution problem is established.