A general Simonenko local principle and Fredholm condition for isotypical components. (English) Zbl 07517408
The article under review is concerned with \(G\)-invariant elliptic pseudodifferential operators which act on sections of vector bundles over a \(G\)-manifold \(M\). It generalizes and expands earlier results by the authors, which characterise the Fredhlmness of operators as such. Explicitely, it is shown in full generality that, when \(G\) is a compact Lie group, \(P \in \Psi^m(M;E_0,E_1)^G\) and \(\alpha\) an irreducible unitary representation of \(G\), then the Fredholmness of \(\pi_{\alpha}(P)\) is equivalent to the transversal \(\alpha\)-ellipticity of \(P\), which in turn is equivalent to the local \(\alpha\)-invertibility of \(P\). This type of result is known as Simonenko’s equivariant localization principle.
Reviewer: Iakovos Androulidakis (Athína)
MSC:
| 47A53 | (Semi-) Fredholm operators; index theories |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 47L80 | Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) |
| 46N20 | Applications of functional analysis to differential and integral equations |
Keywords:
Fredholm operator; \(C^{\ast}\)-algebra; pseudodifferential operator; group actions; induced representationsReferences:
| [1] | Allan, GR, Ideals of vector-valued functions, Proc. Lond. Math. Soc., 3, 2, 193-216 (1968) · Zbl 0194.44501 |
| [2] | Androulidakis, I., Mohsen, O., Yuncken, R.: The convolution algebra of schwarz kernels on a singular foliation. arXiv preprintarXiv:1910.02623 (2019) |
| [3] | Androulidakis, I.; Skandalis, G., Pseudodifferential calculus on a singular foliation, J. Noncommut. Geom., 5, 1, 125-152 (2011) · Zbl 1216.53029 |
| [4] | Atiyah, M.: Elliptic operators and compact groups. Lecture Notes in Mathematics, vol. 401. Springer-Verlag, Berlin-New York, (1974) · Zbl 0297.58009 |
| [5] | Atiyah, M.F., Singer, I.M.: The index of elliptic operators I. Ann. Math. 484-530, (1968) · Zbl 0164.24001 |
| [6] | Baldare, A., The index of \({G}\)-transversally elliptic families. I, J. Noncommut. Geom., 14, 3, 1129-1169 (2020) · Zbl 1466.19007 |
| [7] | Baldare, A., The index of \({G}\)-transversally elliptic families. II, J. Noncommut. Geom., 14, 3, 1171-1207 (2020) · Zbl 1466.19008 |
| [8] | Baldare, A.; Benameur, M-T, The index of leafwise \({G}\)-transversally elliptic operators on foliations, J. Geom. Phys., 163, 104128 (2021) · Zbl 1461.19003 |
| [9] | Baldare, A.; Côme, R.; Lesch, M.; Nistor, V., Fredholm conditions for invariant operators: finite abelian groups and boundary valuel problems, J. Operator Theory, 85, 229-256 (2019) · Zbl 07606459 |
| [10] | Baldare, A., Côme, R., Lesch, M., Nistor, V.: Fredholm conditions for restrictions of invariant pseudodifferential to isotypical components. To appear in Muenster J. Mathematics, Accepted March (2021) · Zbl 1483.58005 |
| [11] | Baldare, A., Côme, R., Nistor, V.: Fredholm conditions for operators invariant with respect to compact Lie group actions. To appear in CRAS, Accepted August (2021) · Zbl 1477.58016 |
| [12] | Berline, N.; Vergne, M., L’indice équivariant des opérateurs transversalement elliptiques, Invent. Math., 124, 1, 51-101 (1996) · Zbl 0883.58037 |
| [13] | Bohlen, K., Boutet de Monvel operators on lie manifolds with boundary, Adv. Math., 312, 234-285 (2017) · Zbl 1365.58015 |
| [14] | Böttcher, A.; Krupnik, N.; Silbermann, B., A general look at local principles with special emphasis on the norm computation aspect, Integr. Equ. Operat. Theory, 11, 4, 455-479 (1988) · Zbl 0655.46036 |
| [15] | Boutet de Monvel, L., Boundary problems for pseudo-differential operators, Acta Math., 126, 1, 11-51 (1971) · Zbl 0206.39401 |
| [16] | Bredon, GE, Introduction to compact transformation groups (1972), UK: Academic press, UK · Zbl 0246.57017 |
| [17] | Brüning, J.; Heintze, E., Representations of compact Lie groups and elliptic operators, Invent. Math., 50, 2, 169-203 (1978) · Zbl 0392.58015 |
| [18] | Brüning, J.; Lesch, M., Hilbert complexes, J. Funct. Anal., 108, 1, 88-132 (1992) · Zbl 0826.46065 |
| [19] | Carrillo Rouse, P.; Lescure, J.-M, Geometric obstructions for Fredholm boundary conditions for manifolds with corners, Ann. K Theory, 3, 3, 523-563 (2018) · Zbl 1394.19003 |
| [20] | Carrillo Rouse, P.; So, BK, K-theory and index theory for some boundary groupoids, RM, 75, 4, 1-20 (2020) · Zbl 1453.58006 |
| [21] | Carvalho, C.; Côme, R.; Qiao, Yu, Gluing action groupoids: Fredholm conditions and layer potentials, Rev. Roumaine Math. Pures Appl., 64, 2-3, 113-156 (2019) · Zbl 1449.35467 |
| [22] | Carvalho, C., Nistor, V., Qiao, Yu: Fredholm conditions on non-compact manifolds: theory and examples. In Operator theory, operator algebras, and matrix theory, vol. 267 of Oper. Theory Adv. Appl., pages 79-122. Birkhäuser/Springer, Cham (2018) · Zbl 1446.58011 |
| [23] | Chandler-Wilde, S., Lindner, M.: Limit operators, collective compactness, and the spectral theory of infinite matrices, vol. 210. American Mathematical Society (2011) · Zbl 1219.47001 |
| [24] | Côme, R., The Fredholm property for groupoids is a local property, RM, 74, 4, 160 (2019) · Zbl 1430.58020 |
| [25] | Connes, A., Noncommutative geometry (1994), San Diego: Academic Press, San Diego · Zbl 0818.46076 |
| [26] | Debord, C.; Skandalis, G., Adiabatic groupoid, crossed product by \(\mathbb{R}_+^\ast\) and pseudodifferential calculus, Adv. Math., 257, 66-91 (2014) · Zbl 1300.58007 |
| [27] | Dixmier, J.: Les \(C^*\)-algèbres et leurs représentations. Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. Éditions Jacques Gabay, Paris, 1996. Reprint of the second (1969) edition |
| [28] | Douglas, RG, Banach algebra techniques in operator theory (2012), UK: Springer Science & Business Media, UK |
| [29] | Favard, J., Sur les équations différentielles linéaires à coefficients presque-périodiques, Acta Math., 51, 1, 31-81 (1928) · JFM 53.0409.02 |
| [30] | Georgescu, V., On the structure of the essential spectrum of elliptic operators on metric spaces, J. Funct. Anal., 260, 6, 1734-1765 (2011) · Zbl 1242.47052 |
| [31] | Hagen, R.; Roch, S.; Silbermann, B., C*-algebras and numerical analysis (2000), USA: CRC Press, USA · Zbl 0964.65055 |
| [32] | Hagger, R., Limit operators, compactness and essential spectra on bounded symmetric domains, J. Math. Anal. Appl., 470, 1, 470-499 (2019) · Zbl 06958722 |
| [33] | Hochs, P., Wang, B.-L., Wang, H.: An equivariant Atiyah-Patodi-Singer index theorem for proper actions. arXiv preprintarXiv:1904.11146 (2019) |
| [34] | Hörmander, L.: The analysis of linear partial differential operators. III. Classics in Mathematics. Springer, Berlin, (2007). Pseudo-differential operators, Reprint of the 1994 edition · Zbl 1115.35005 |
| [35] | Julg, P., Induction holomorphe pour le produit croisé d’une C*-algebre par un groupe de Lie compact, CR Acad. Sci. Paris Sér. I Math, 294, 5, 193-196 (1982) · Zbl 0516.46047 |
| [36] | Kasparov, G., Elliptic and transversally elliptic index theory from the viewpoint of KK-theory, J. Noncommut. Geom., 10, 4, 1303-1378 (2016) · Zbl 1358.19001 |
| [37] | Kisil, VV, Operator covariant transform and local principle, J. Phys. A Math. Theor., 45, 24, 244022 (2012) · Zbl 1298.81111 |
| [38] | Lauter, R.; Monthubert, B.; Nistor, V., Pseudodifferential analysis on continuous family groupoids, Doc. Math., 5, 625-657 (2000) · Zbl 0961.22005 |
| [39] | Lescure, J-M; Manchon, D.; Vassout, S., About the convolution of distributions on groupoids, J. Noncommut. Geom., 11, 2, 757-789 (2017) · Zbl 1385.46026 |
| [40] | Lindner, M.; Seidel, M., An affirmative answer to a core issue on limit operators, J. Funct. Anal., 267, 3, 901-917 (2014) · Zbl 1292.47020 |
| [41] | Măntoiu, M., Essential spectrum and Fredholm properties for operators on locally compact groups, J. Operator Theory, 77, 2, 481-501 (2017) · Zbl 1449.46058 |
| [42] | Măntoiu, M., Nistor, V.: Spectral theory in a twisted groupoid setting: spectral decompositions, localization and Fredholmness. Münster J. of Math., 13 (2020) · Zbl 1457.58019 |
| [43] | Nistor, V., Prudhon, N.: Exhausting families of representations and spectra of pseudodifferential operators. arXiv preprintarXiv:1411.7921, (2014) · Zbl 1424.46073 |
| [44] | Paradan, P.-É., Vergne, M.: Index of transversally elliptic operators. Astérisque, in honor of the 60th birthday of J.-M. Bismut(328):297-338 (2010) · Zbl 1209.19003 |
| [45] | Rabinovich, V., Roch, S., Silbermann, B.: Limit operators and their applications in operator theory, volume 150 of Operator Theory: Advances and Applications. Birkhäuser, (2004) · Zbl 1077.47002 |
| [46] | Rabinovich, V.; Schulze, B.-W; Tarkhanov, N., C*-algebras of singular integral operators in domains with oscillating conical singularities, Manuscr. Math., 108, 1, 69-90 (2002) · Zbl 1063.47040 |
| [47] | Roch, S.; Santos, PA; Silbermann, B., Non-commutative Gelfand theories: a tool-kit for operator theorists and numerical analysts (2010), UK: Springer Science & Business Media, UK · Zbl 1209.47002 |
| [48] | Savin, A.; Schrohe, E., Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations, J. Funct. Anal., 278, 5, 108400 (2020) · Zbl 1469.58015 |
| [49] | Schrohe, E., Fréchet algebra techniques for boundary value problems on noncompact manifolds: Fredholm criteria and functional calculus via spectral invariance, Math. Nachr., 199, 1, 145-185 (1999) · Zbl 0923.58054 |
| [50] | Schulze, BW, An algebra of boundary value problems not requiring Shapiro-Lopatinskij conditions, J. Funct. Anal., 179, 2, 374-408 (2001) · Zbl 0984.58013 |
| [51] | Semenyuta, V.N., Khevelev, A.V.: A local principle for special classes of Banach algebras. Izv. Severo-Kavkazskogo Nauchn. Tsentra Vyssh. Shkoly Ser. Estestv. Nauk, 1:15-17, (1977) |
| [52] | Simonenko, IB, A new general method for investigating linear operator equations of the singular integral operator type. I, II, Izv. Akad. Nauk SSSR Ser. Mat, 29, 567-586 (1965) · Zbl 0146.13101 |
| [53] | Simonenko, I.B., Min, C. N.: The local method in the theory of one-dimensional singular integral equations with piecewise continuous coefficients. Noetherity.(Russian) Rostov University Press, Rostov, (1986) |
| [54] | Singer, I.M.: Recent applications of index theory for elliptic operators. In Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), pp. 11-31, (1973) · Zbl 0293.58013 |
| [55] | Sternin, BY; Shatalov, VE; Schulze, BW, On general boundary-value problems for elliptic equations, Sbornik Math., 189, 10, 1573 (1998) · Zbl 0926.35033 |
| [56] | tom Dieck, T.: Transformation groups, volume 8 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin (1987) · Zbl 0611.57002 |
| [57] | Trèves, F.: Introduction to pseudodifferential and Fourier integral operators. Vol. 1. Plenum Press, New York-London, 1980. Pseudodifferential operators, The University Series in Mathematics · Zbl 0453.47027 |
| [58] | Vasilyev, V.B.: Pseudo-differential operators on manifolds with a singular boundary. In: Modern Problems in Applied Analysis, pp. 169-179. Springer, (2018) · Zbl 1420.58010 |
| [59] | Xia, J., A double commutant relation in the Calkin algebra on the Bergman space, J. Funct. Anal., 274, 6, 1631-1656 (2018) · Zbl 1394.46044 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
