Toeplitz operators on bounded symmetric domains
HTML articles powered by AMS MathViewer
- by Harald Upmeier
- Trans. Amer. Math. Soc. 280 (1983), 221-237
- DOI: https://doi.org/10.1090/S0002-9947-1983-0712257-2
- PDF | Request permission
Abstract:
In this paper Jordan algebraic methods are applied to study Toeplitz operators on the Hardy space ${H^2}(S)$ associated with the Shilov boundary $S$ of a bounded symmetric domain $D$ in ${{\mathbf {C}}^n}$ of arbitrary rank. The Jordan triple system $Z \approx {{\mathbf {C}}^n}$ associated with $D$ is used to determine the relationship between Toeplitz operators and differential operators. Further, it is shown that each Jordan triple idempotent $e \in Z$ induces an irreducible representation ("$e$-symbol") of the ${C^{\ast } }$-algebra $\mathcal {T}$ generated by all Toeplitz operators ${T_f}$ with continuous symbol function $f$.References
- C. A. Berger and L. A. Coburn, Wiener-Hopf operators on $U_{2}$, Integral Equations Operator Theory 2 (1979), no. 2, 139–173. MR 543881, DOI 10.1007/BF01682732
- Charles A. Berger, Lewis A. Coburn, and Adam Korányi, Opérateurs de Wiener-Hopf sur les sphères de Lie, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 21, A989–A991 (French, with English summary). MR 584284
- L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators, Annals of Mathematics Studies, vol. 99, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1981. MR 620794, DOI 10.1515/9781400881444
- Louis Boutet de Monvel, On the index of Toeplitz operators of several complex variables, Invent. Math. 50 (1978/79), no. 3, 249–272. MR 520928, DOI 10.1007/BF01410080
- Hel Braun and Max Koecher, Jordan-Algebren, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 128, Springer-Verlag, Berlin-New York, 1966 (German). MR 0204470
- Robert Braun, Wilhelm Kaup, and Harald Upmeier, A holomorphic characterization of Jordan $C^*$-algebras, Math. Z. 161 (1978), no. 3, 277–290. MR 493373, DOI 10.1007/BF01214510
- L. A. Coburn, Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J. 23 (1973/74), 433–439. MR 322595, DOI 10.1512/iumj.1973.23.23036
- Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
- Ronald G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, Vol. 49, Academic Press, New York-London, 1972. MR 0361893
- Alexander Dynin, Inversion problem for singular integral operators: $C^{\ast }$-approach, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 10, 4668–4670. MR 507929, DOI 10.1073/pnas.75.10.4668
- Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
- L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by Leo Ebner and Adam Korányi. MR 0171936
- J. Janas, Toeplitz operators related to certain domains in $C^{n}$, Studia Math. 54 (1975), no. 1, 73–79. MR 390827, DOI 10.4064/sm-54-1-73-79
- Nicholas P. Jewell and Steven G. Krantz, Toeplitz operators and related function algebras on certain pseudoconvex domains, Trans. Amer. Math. Soc. 252 (1979), 297–312. MR 534123, DOI 10.1090/S0002-9947-1979-0534123-7
- Kenneth D. Johnson, On a ring of invariant polynomials on a Hermitian symmetric space, J. Algebra 67 (1980), no. 1, 72–81. MR 595020, DOI 10.1016/0021-8693(80)90308-7
- Max Koecher, An elementary approach to bounded symmetric domains, Rice University, Houston, Tex., 1969. MR 0261032
- Adam Korányi, The Poisson integral for generalized half-planes and bounded symmetric domains, Ann. of Math. (2) 82 (1965), 332–350. MR 200478, DOI 10.2307/1970645
- Daniel A. Levine, Systems of singular integral operators on spheres, Trans. Amer. Math. Soc. 144 (1969), 493–522. MR 412743, DOI 10.1090/S0002-9947-1969-0412743-1
- Ottmar Loos, Jordan pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. MR 0444721 —, Bounded symmetric domains and Jordan pairs, Univ. of California, Irvine, 1977.
- Paul S. Muhly and Jean N. Renault, $C^{\ast }$-algebras of multivariable Wiener-Hopf operators, Trans. Amer. Math. Soc. 274 (1982), no. 1, 1–44. MR 670916, DOI 10.1090/S0002-9947-1982-0670916-3
- Raghavan Narasimhan, Several complex variables, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1971. MR 0342725
- S. C. Power, Commutator ideals and pseudodifferential $C^{\ast }$-algebras, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 124, 467–489. MR 596980, DOI 10.1093/qmath/31.4.467
- Iain Raeburn, On Toeplitz operators associated with strongly pseudoconvex domains, Studia Math. 63 (1978), no. 3, 253–258. MR 515494, DOI 10.4064/sm-63-3-253-258
- Wilfried Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/70), 61–80 (German). MR 259164, DOI 10.1007/BF01389889
- Masaru Takeuchi, Polynomial representations associated with symmetric bounded domains, Osaka Math. J. 10 (1973), 441–475. MR 412493
- Harald Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math. 108 (1986), no. 1, 1–25 (1986). MR 821311, DOI 10.2307/2374466
- U. Venugopalkrishna, Fredholm operators associated with strongly pseudoconvex domains in $C^{n}$, J. Functional Analysis 9 (1972), 349–373. MR 0315502, DOI 10.1016/0022-1236(72)90007-9
- Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Band 188, Springer-Verlag, New York-Heidelberg, 1972. MR 0498999
Bibliographic Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 280 (1983), 221-237
- MSC: Primary 47B35; Secondary 32M15, 46L99
- DOI: https://doi.org/10.1090/S0002-9947-1983-0712257-2
- MathSciNet review: 712257