TY - ICOMM

T1 - Perturbation of Sectorial Projections of Elliptic

AU - Booss-Bavnbek, Bernhelm

AU - Chen, Guoyuan

AU - Lesch, Matthias

AU - Zhu, Chaofeng

N1 - To be published in peer-reviewed learned journal

PY - 2010/12/29

Y1 - 2010/12/29

N2 - 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 ξ\ne 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.

AB - 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 ξ\ne 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.

M3 - Net publication - Internet publication

ER -