Toeplitz operators on bounded symmetric domains

Upmeier, Harald

doi:10.1090/S0002-9947-1983-0712257-2

Toeplitz operators on bounded symmetric domains
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Trans. Amer. Math. Soc. 280 (1983), 221-237 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

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