Toeplitz operators and weighted Wiener-Hopf operators, pseudoconvex Reinhardt and tube domains
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- by Norberto Salinas PDF
- Trans. Amer. Math. Soc. 336 (1993), 675-699 Request permission
Abstract:
The notion of weighted Wiener-Hopf operators is introduced. Their relationship with Toeplitz operators acting on the space of holomorphic functions which are square integrable with respect to a given "symmetric" measure is discussed. The groupoid approach is used in order to present a general program for studying the ${C^{\ast } }$-algebra generated by weighted Wiener-Hopf operators associated with a solid cone of a second countable locally compact Hausdorff group. This is applied to the case when the group is the dual of a connected locally compact abelian Lie group and the measure is "well behaved" in order to produce a geometric groupoid which is independent of the representation. The notion of a Reinhardt-tube domain $\Omega$ appears thus naturally, and a decomposition series of the corresponding ${C^{\ast } }$-algebra is presented in terms of groupoid ${C^{\ast } }$-algebras associated with various parts of the boundary of the domain $\Omega$.References
- Eric Bedford and Jiri Dadok, Generalized Reinhardt domains, J. Geom. Anal. 1 (1991), no. 1, 1–17. MR 1097933, DOI 10.1007/BF02938112
- Raul E. Curto, Fredholm and invertible $n$-tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1981), no. 1, 129–159. MR 613789, DOI 10.1090/S0002-9947-1981-0613789-6 —, Reinhardt domains and operator theory, Proc. Sympos. Pure Math., vol. 52, part 3, Amer. Math. Soc., Providence, R. I., 1990, pp. 93-101.
- Raúl E. Curto and Paul S. Muhly, $C^\ast$-algebras of multiplication operators on Bergman spaces, J. Funct. Anal. 64 (1985), no. 3, 315–329. MR 813203, DOI 10.1016/0022-1236(85)90062-X
- Raúl E. Curto and Norberto Salinas, Spectral properties of cyclic subnormal $m$-tuples, Amer. J. Math. 107 (1985), no. 1, 113–138. MR 778091, DOI 10.2307/2374459
- 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
- Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639 —, The analysis of linear partial differential operators, Springer-Verlag, New York, 1983.
- Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
- Qing Lin and Norberto Salinas, Proper holomorphic maps and analytic Toeplitz $n$-tuples, Indiana Univ. Math. J. 39 (1990), no. 3, 547–562. MR 1078730, DOI 10.1512/iumj.1990.39.39030
- 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
- Alexandru Nica, Some remarks on the groupoid approach to Wiener-Hopf operators, J. Operator Theory 18 (1987), no. 1, 163–198. MR 912819
- Efton Park, Index theory and Toeplitz algebras on certain cones in $\textbf {Z}^2$, J. Operator Theory 23 (1990), no. 1, 125–146. MR 1054820
- Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
- R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR 847923, DOI 10.1007/978-1-4757-1918-5
- Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266
- Norberto Salinas, The $\overline \partial$-formalism and the $C^*$-algebra of the Bergman $n$-tuple, J. Operator Theory 22 (1989), no. 2, 325–343. MR 1043731 —, Non-compactness of the $\overline \partial$-Neumann problem and Toeplitz ${C^{\ast } }$-algebras, Proc. Sympos. Pure Math., vol. 52, part 3, Amer. Math. Soc., Providence, R. I., 1991, pp. 329-334.
- Norberto Salinas, Albert Sheu, and Harald Upmeier, Toeplitz operators on pseudoconvex domains and foliation $C^*$-algebras, Ann. of Math. (2) 130 (1989), no. 3, 531–565. MR 1025166, DOI 10.2307/1971454 N. Salinas and H. Upmeier, Holomorphic foliations and Toeplitz ${C^{\ast } }$-algebras (in preparation). A. G. Sergeev, On matrix Reinhardt domains, preprint, 1989.
- Albert Jeu-Liang Sheu, Isomorphism of the Toeplitz $C^*$-algebras for the Hardy and Bergman spaces on certain Reinhardt domains, Proc. Amer. Math. Soc. 116 (1992), no. 1, 113–120. MR 1092926, DOI 10.1090/S0002-9939-1992-1092926-9
- Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR 0361899
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 336 (1993), 675-699
- MSC: Primary 47B35; Secondary 32A07, 46L05, 47B38
- DOI: https://doi.org/10.1090/S0002-9947-1993-1093217-8
- MathSciNet review: 1093217