Toeplitz operators and weighted Wiener-Hopf operators, pseudoconvex Reinhardt and tube domains

Author:
Norberto Salinas

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

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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 -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 appears thus naturally, and a decomposition series of the corresponding -algebra is presented in terms of groupoid -algebras associated with various parts of the boundary of the domain .

**[1]**E. Bedford and J. Dadok,*Generalized Reinhardt domains*, preprint, 1990. MR**1097933 (91m:32034)****[2]**R. Curto,*Fredholm and invertible tuples of operators, the deformation problem*, Trans. Amer. Math. Soc.**226**(1981), 129-159. MR**613789 (82g:47010)****[3]**-,*Reinhardt domains and operator theory*, Proc. Sympos. Pure Math., vol. 52, part 3, Amer. Math. Soc., Providence, R. I., 1990, pp. 93-101.**[4]**R. Curto and P. Muhly, -*algebras of multiplication operators on Bergman spaces*, J. Funct. Anal.**61**(1985), 315-329. MR**813203 (86m:47044)****[5]**R. Curto and N. Salinas,*Spectral properties of cyclic subnormal*-*tuples*, Amer. J. Math.**107**(1985), 113-138. MR**778091 (86g:47024)****[6]**J. Dixmier, -*algebras and their representations*, North-Holland, Amsterdam, 1977. MR**0458185 (56:16388)****[7]**L. Hormander,*Introduction to complex analysis in several variables*, North-Holland, Amsterdam, 1973. MR**1045639 (91a:32001)****[8]**-,*The analysis of linear partial differential operators*, Springer-Verlag, New York, 1983.**[9]**S. Krantz,*Function theory of several complex variables*, Wiley, New York, 1982. MR**635928 (84c:32001)****[10]**Q. Lin and N. Salinas,*Proper holomorphic map and analytic Toeplitz*-*tuples*, Indiana Univ. Math. J.**39**(1990), 547-562. MR**1078730 (92e:47042)****[11]**P. Muhly and J. Renault, -*algebras of multivariable Wiener-Hopf operators*, Trans. Amer. Math. Soc.**274**(1982), 1-44. MR**670916 (84h:46074)****[12]**A. Nica,*Some remarks on the groupoid approach to Wiener-Hopf operators*, J. Operator Theory**18**(1987), 163-198. MR**912819 (89d:47045)****[13]**E. Park,*Index theory and Toeplitz algebras on certain cones in*, J. Operator Theory**23**(1990), 125-146. MR**1054820 (91i:47039)****[14]**G. Pedersen, -*algebras and their automorphism groups*, London Math. Soc. Monographs, vol. 10, Academic Press, London, 1979. MR**548006 (81e:46037)****[15]**M. Range,*Holomorphic functions and integral representation in several complex variables*, Springer-Verlag, Berlin and New York, 1986. MR**847923 (87i:32001)****[16]**J. Renault,*A groupoid approach to*-*algebras*, Lecture Notes in Math., vol. 793, Springer-Verlag, New York, 1980. MR**584266 (82h:46075)****[17]**N. Salinas,*The*-*formalism and the*-*algebra of the Bergman*-*tuple*, J. Operator Theory**22**(1989), 325-343. MR**1043731 (91b:47054)****[18]**-,*Non-compactness of the*-*Neumann problem and Toeplitz*-*algebras*, Proc. Sympos. Pure Math., vol. 52, part 3, Amer. Math. Soc., Providence, R. I., 1991, pp. 329-334.**[19]**N. Salinas, A. Sheu, and H. Upmeier,*Toeplitz operators on pseudoconvex domains and foliation*-*algebras*, Ann. of Math.**130**(1989), 531-565. MR**1025166 (91e:47026)****[20]**N. Salinas and H. Upmeier,*Holomorphic foliations and Toeplitz*-*algebras*(in preparation).**[21]**A. G. Sergeev,*On matrix Reinhardt domains*, preprint, 1989.**[22]**A. Sheu,*On the isomorphism between Toeplitz*-*algebras of the Hardy and the Bergman spaces*, Proc. Amer. Math. Soc.**116**(1992), 113-120. MR**1092926 (92k:47054)****[23]**A. Shields,*Weighted shift operators and analytic function theory*, Math. Surveys Monographs, vol. 13, Amer. Math. Soc., Providence, R. I., 1974, pp. 49-128. MR**0361899 (50:14341)**

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DOI:
https://doi.org/10.1090/S0002-9947-1993-1093217-8

Article copyright:
© Copyright 1993
American Mathematical Society