×

zbMATH — the first resource for mathematics

Zur Spektralinvarianz von Algebren von Pseudodifferentialoperatoren in der \(L^ p\)-Theorie. (The spectral invariance of algebras of pseudodifferential operators in the \(L^ p\)-theory). (German) Zbl 0674.47033
Let \(\Psi^ 0_{1,\delta}(0\leq \delta <1)\) be the Hörmander class of pseudodifferential operators. It is shown that if an operator P from \(\Psi^ 0_{1,\delta}\) is invertible in \({\mathfrak L}(L^ p({\mathbb{R}}^ n))\) then \(P^{-1}\in \Psi^ 0_{1,\delta}\). This statement means, in particular, that \(\Psi^ 0_{1,\delta}\) is a so-called \(\Psi\)-algebra in \({\mathfrak L}(L^ p({\mathbb{R}}^ n))\). The proof is based on methods developed earlier by R. Beals, R. R. Coifman, Y. Meyer and H. O. Cordes.
Reviewer: J.Banas

MSC:
47Gxx Integral, integro-differential, and pseudodifferential operators
35S05 Pseudodifferential operators as generalizations of partial differential operators
46H05 General theory of topological algebras
46K05 General theory of topological algebras with involution
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Adams, R.:Sobolev Spaces, Academic Press New York, San Francisco, London (1975) · Zbl 0314.46030
[2] Beals, R.: Characterisation of pseudodifferential operators and applications,Duke Math. J. 44 (1977), S. 45–57; ibid.46 (1979), S. 215 · Zbl 0353.35088
[3] Beals, R.: On the boundedness of pseudodifferential operators,Comm. in part. diff. eq. 2 (10) (1977), S. 1063–1070 · Zbl 0397.35072
[4] Beals, R.: L p -and Hölder-estimates for pseudodifferential operators,Ann. Inst. Fourier 29,3 (1979), S. 239–260 · Zbl 0387.35065
[5] Beals, R.: Weighted distribution spaces and pseudodifferential operators,Journal d’Analyse 39 (1981), S. 131–187 · Zbl 0474.35089
[6] Bratteli, O.:Derivations, Dissipations and Group Actions on C *-Algebras, Lecture Notes in Mathematics 1229, Springer Verlag Berlin, Heidelberg, New York, Tokyo (1986) · Zbl 0607.46035
[7] Bratteli, O., Elliot, G. A., Jørgensen, P. E. T.: Decomposition of unbounded derivations into invariant and approximately inner parts,Journal für reine und angewandte Mathematik 346 (1984), S. 166–193 · Zbl 0515.46057
[8] Calderón, A. P., Vaillancourt, R.: On the boundedness of pseudo-differential operators,J. Math. Soc. Japan 23 (1971), S. 374–378 · Zbl 0214.39004
[9] Calderón, A. P., Vaillancourt, R.: A class of bounded pseudo-differential operators,Proc. Nat. Acad. Sci. USA (1972), S. 1185–1187 · Zbl 0244.35074
[10] Coifman, R., Meyer, Y.:Au delà des opérateurs pseudo-différentiels, Astérisques57 (1978)
[11] Connes, A.: C*-algèbres et géometrie différentielle,C. R. Acad. Sci. Paris 290,Ser. A (1980), S. 599–604 · Zbl 0433.46057
[12] Connes, A.: An analogue of the Thom-isomorphism for crossed products of a C*-algebra by an action of \(\mathbb{R}\),Advances in Math. 39 (1981), S. 31–55 · Zbl 0461.46043
[13] Cordes, H. O.: On compactness of commutators of multiplications and convolutions and boundedness of pseudodifferential operators,J. Functional Analysis 18 (1975), S. 115–131 · Zbl 0306.47024
[14] Cordes, H. O.: On pseudodifferential operators and smoothness of special Lie-Group Representations,Manuscripta Mathematica 28 (1979), S. 51–69 · Zbl 0415.35083
[15] Cordes, H. O.: C*-Algebras and Fréchet*-Algebras,Proc. Symp. Pure Math. AMS 43 (1985), S. 79–104
[16] Dunau, J.: Fonctions d’un opérateur elliptique sur une variété compacte,J. Math. Pure Appl. 56 (1977), S. 367–391 · Zbl 0323.58017
[17] Fefferman, C: L p -bounds for pseudo-differential operators,Israel J. Math. 14 (1973), S. 413–417 · Zbl 0259.47045
[18] Gramsch, B.: Relative Inversion in der Störungstheorie von Operatoren und {\(\Psi\)}-Algebren,Math. Annalen 269 (1984), S. 27–71 · Zbl 0661.47037
[19] Gramsch, B.:Operatoralgebren I und II, Vorlesungen an der Universität Mainz (1985/86) (unveröffentlicht)
[20] Gramsch, B., Kalb, G.:Pseudo-Locality and Hypoellipticity in Operator Algebras, Semesterbericht Funktionalanalysis, Tübingen Sommersemester 1985
[21] Hörmander, L.: Pseudo-differential operators and hypoelliptic equations,Proc. Symp. Pure Math. AMS 10 (1967), S. 138–183 · Zbl 0167.09603
[22] Illner, R.: A class of L p -bounded pseudo-differential operators,Proc. Amer. Math. Soc. 51 (1975), S. 347–355 · Zbl 0279.47017
[23] Kato, T.: Boundedness of some pseudo-differential operators,Osaka J. Math. 13 (1976), S. 1–9 · Zbl 0342.47029
[24] Kumano-go, H.:Pseudo-differential operators, MIT-Press (1982) · Zbl 0489.35003
[25] Nagase, M.: The L p -boundedness of pseudo-differential operators with nonregular symbols,Comm. in Part. Diff. Eq. 2 (10) (1977), S. 1045–1061 · Zbl 0397.35071
[26] Schrohe, E.: A {\(\Psi\)}*-algebra of pseudodifferential operators on noncompact manifoldsArch. Math. 49 (1988) (erscheint demnächst) · Zbl 0631.47035
[27] Ueberberg, J.:Fréchetalgebren mit Spektralinvarianz in der L p -Theorie der Pseudodifferentialoperatoren nach Ergebnissen von R. Beals, Diplomarbeit am Fachbereich Mathematik der Universität Mainz (1987)
[28] Wagner, K.:Kommutatoren in der Theorie der Pseudodifferentialoperatoren mit Anwendungen auf die Submultiplikativität der Klassen S 1/2, 1/2 0 , Diplomarbeit am Fachbereich Mathematik der Universität Mainz (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.