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In addition we guarantee that the resulting block matrix operators locally belong to B μ,d (R+ ; k, k; j− , j+ ) and smoothly depend on (y, η), for k = dim Ey = dim Fy , j± = dim J±,y . The corresponding operator functions k(y, η), t(y, η) and q(y, η) can be extended from S ∗ Y to T ∗ Y \ 0 by κλ -homogeneity of order μ. 25). Then applying an operator convention which assigns to such principal symbols associated operators gives us the additional entries. 9. 16) is satisfied. It turns out that, from the point of view of the associated Fredholm indices, for every two such A1 , A2 with the same upper left corner we can construct an elliptic operator R on the boundary such that indA1 − indA2 = indR.

Let a(τ ) ∈ Scl (R) be elliptic in the sense L(a) ⊂ C\ {0}. 2) is a Fredholm operator if and only if 0 ∈ M (a). 5) The union C(a) := L(a) ∪ M (a) is a continuous and piecewise smooth curve which can be represented as the image under a continuous map γ : [0, 1] → C. 4) holds, we have a winding number wind C(a), and there is the well-known relation ind op+ (a) = wind C(a). 2) cannot be Fredholm in this case. 3. 6) j = −aj for all j ∈ N. μ (R) denote the space of all symbols with the anti-transmission property.

W. Schulze, Pseudo-Differential Boundary Value Problems, Conical Singularities, and Asymptotics, Akademie Verlag, 1994. -W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, Wiley, 1998. -W. Schulze, Operator algebras with symbol hierarchies on manifolds with singularities, in Advances in Partial Differential Equations (Approaches to Singular Analysis), Editors: J. Gil, D. Grieser and M. Lesch, Operator Theory: Advances and Applications Birkh¨ auser, 2001, 167–207. -W. Schulze, An algebra of boundary value problems not requiring Shapiro– Lopatinskij conditions, J.

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