Abstract
Under a suitable ellipticity condition, we show that classical SG-pseudodifferential operators of nonnegative order possess complex powers. We show that the powers are again classical and derive an explicit formula for all homogeneous components.
Keywords: Complex power, Weighted symbols, Noncompact manifolds
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
1. Ammann, B., Lauter, R., Nistor, V., Vasy, A.: Complex powers and non-compact manifolds. Comm. Partial Differential Equations 29, 671-705 (2004)
2. Buzano, E., Nicola, F.: Complex powers of hypoelliptic pseudodifferential operators. Preprint Università di Torino (2005)
3. Bucicovschi, B.: An extension of the work of V. Guillemin on complex powers and zeta functions of elliptic pseudodifferential operators. Proc. Amer. Math. Soc. 127, 3081-3090 (1999)
4. Cordes, H.O.: The Technique of Pseudodifferential Operators over Rn. Univ. Press, (1995)
5. Coriasco, S., Maniccia, L.: Wave front set at infinity and hyperbolic linear operators with multiple characteristics. Ann. Global Anal. Geom. 24, 375-400 (2003)
6. Coriasco, S., Panarese, P.: Fourier integral operators defined by classical symbols with exit behaviour. Math.Nachr. 242, 61-78 (2002)
7. Egorov, V.V., Schulze, B.W.: Pseudo-differential Operators, Singularities, Applications. Operator Theory: Advances and Applications, 93, Birkhäuser, (1997)
8. Grubb, G., Hansen, L.: Complex powers of resolvents of pseudodifferential operators. Comm. Partial Differential Equations 27, 2333-2361 (2002)
9. Grubb, G.: Complex powers of pseudodifferential boundary value problems with the transmission property. Lecture Notes in Math.: Pseudodifferential Operators, 1256, Springer-Verlag, Berlin (1987)
10. Guillemin, V.: A new proof of Weyl's formula on the asymptotic distribution of eigenvalues. Adv. in Math. 55, 131-160 (1985)
11. Hirschmann, T.: Pseudo-differential operators and asymptotics on manifolds with corners V, Va. Preprint Karl-Weierstrass-Institut für Math., Berlin (1991)
12. Kumano-go, H.: Pseudo-differential Operators. The MIT Press, (1981)
13. Loya, P.: Complex powers of differential operators on manifolds with conical singularities. J. Anal. Math. 89, 31-56 (2003)
14. Maniccia, L., Panarese, P.: Eigenvalue asymptotics for a class of md-elliptic ψdo's on manifolds with cylindrical exits. Ann. Mat. Pura Appl. 181, no. 3, 283-308 (2002)
15. Nicola, F.: Trace functionals for a class of pseudo-differential operators in Rn. Math. Phys. Anal. Geom. 6, 89-105 (2003)
16. Parenti, C.: Operatori pseudodifferenziali in Rn e applicazioni. Ann. Mat. Pura Appl. 93, 359-389 (1972)
17. Rempel, S., Schulze, B.W.: Complex powers for pseudodifferential boundary value problems I, II. Math. Nachr. 111, 41-109 (1983); Math. Nachr. 116, 269-314 (1984)
18. Schrohe, E.: Spaces of weighted symbols and weighted Sobolev spaces on manifolds. Pseudodifferential operators, Oberwolfach, (1986), 360-377, Lecture Notes in Math., 1256, Springer Berlin, Berlin (1987)
19. Schrohe, E.: Complex powers of elliptic pseudodifferential operators. Int. Eq. Oper. Th. 9, 337-354 (1986)
20. Schrohe, E.: Complex powers on noncompact manifolds and manifolds with singularities. Math. Ann. 281, 393-409 (1988)
21. Seeley, R.: Complex powers of an elliptic operator. AMS Proc. Symp. Pure Math. 10, 288-307 (1967)
22. Shubin, M.A.: Pseudodifferential Operators and Spectral theory. Springer-Verlag, Berlin (1987)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Maniccia, L., Schrohe, E. & Seiler, J. Complex powers of classical SG-pseudodifferential operators. Ann. Univ. Ferrara 52, 353–369 (2006). https://doi.org/10.1007/s11565-006-0026-1
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11565-006-0026-1