# Perturbation of Sectorial Projections of Elliptic

Bernhelm Booss-Bavnbek, Guoyuan Chen, Matthias Lesch, Chaofeng Zhu

Publikation: AndetUdgivelser på nettet - Net-publikationForskningpeer review

### Resumé

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.
Originalsprog Engelsk 29 dec. 2010 Udgivet - 29 dec. 2010

### Bibliografisk note

To be published in peer-reviewed learned journal

### Citer dette

Booss-Bavnbek, B., Chen, G., Lesch, M., & Zhu, C. (2010, dec 29). Perturbation of Sectorial Projections of Elliptic.
Booss-Bavnbek, Bernhelm ; Chen, Guoyuan ; Lesch, Matthias ; Zhu, Chaofeng. / Perturbation of Sectorial Projections of Elliptic. 2010.
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title = "Perturbation of Sectorial Projections of Elliptic",
abstract = "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.",
author = "Bernhelm Booss-Bavnbek and Guoyuan Chen and Matthias Lesch and Chaofeng Zhu",
note = "To be published in peer-reviewed learned journal",
year = "2010",
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Perturbation of Sectorial Projections of Elliptic. / Booss-Bavnbek, Bernhelm; Chen, Guoyuan; Lesch, Matthias; Zhu, Chaofeng.

2010, .

Publikation: AndetUdgivelser på nettet - Net-publikationForskningpeer review

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 -