In the present chapter we turn our attention to the set M\W of the symbol space M of a general comparison algebra, consisting of all points not contained in the wave front space. As pointed out before, the wave front space W (i.e. the symbol space of the minimal comparison algebra) is found naturally imbedded as an open subset of the symbol space M, for every comparison algebra C meeting our general assumptions. This now will be discussed in detail in sec.1, below. If the manifold Ω is compact, so that the minimal comparison algebra is the only comparison algebra, then we have M=W. In that case every differential expression on Ω is within comparison reach of the algebra J0, as was shown in VI,3. For an elliptic expression L then the realization Z of V, def.6.2 has the property that Z–λ=(A–λ∧N)∧−N is Fredholm for every λ∈C. In other words, there is no essential spectrum, in the sense of [CHe1]). This fact is in direct relation to the fact that the set M\W is void. Essentially it follows that the latter set, or, rather its interior Ms = M\(Wclos), is the origin of the essential spectrum of elliptic operators within reach of a comparison algebra C, (assuming that E =K(H)).

Accordingly we now engage in a detailed discussion of the set Ms, which is called the secondary symbol space of the algebra C.