AbstractsMathematics

Let L_{sc}^m(M,E) denote the space of semi-classical pseudo-differential operators of order m, acting between sections of a Hermitian vector bundle E over a closed Riemannian manifold M . Let A in L_{sc}^m(M,E) be elliptic with principal symbol a_m and m> 0 . We assume that there exist two rays L_{alpha_j}, j = 1, 2 with spec(a_m(x,xi)) cap L_{alpha_j} = emptyset for all x in M and all cotangent vectors ξne 0 . We choose an arc around zero connecting the two rays and making a path Gamma_+ such that spec(A) cap Gamma_+ = emptyset, as well. Then the sectorial projection P_{Gamma_+}(A) is a well-defined bounded operator on the Sobolev spaces H^s(M;E), s in R . We show that P_{Gamma_+}(A) varies continuously as bounded operator in H^s(M;E), if A is continuously varying in a specific sense, depending on a strong topology of the leading symbol and a weaker topology of the lower order parts.