Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Towards a functional calculus for subnormal tuples: the minimal normal extension


Author: John B. Conway
Journal: Trans. Amer. Math. Soc. 326 (1991), 543-567
MSC: Primary 47B20; Secondary 32E30, 47A20, 47A60
DOI: https://doi.org/10.1090/S0002-9947-1991-1005077-X
MathSciNet review: 1005077
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper the study of a functional calculus for subnormal $ n$-tuples is initiated and the minimal normal extension problem for this functional calculus is explored. This problem is shown to be equivalent to a mean approximation problem in several complex variables which is solved. An analogous uniform approximation problem is also explored. In addition these general results are applied together with The Area and the The Coarea Formula from Geometric Measure Theory to operators on Bergman spaces and to the tensor product of two subnormal operators. The minimal normal extension of the tensor product of the Bergman shift with itself is completely determined.


References [Enhancements On Off] (What's this?)

  • [1] M. B. Abrahamse and T. L. Kriete, The spectral multiplicity of a multiplication operator, Indiana Univ. Math. J. 22 (1973), 845-857. MR 0320797 (47:9331)
  • [2] W. B. Arveson, An invitation to $ {C^{\ast}}$-algebra, Springer-Verlag, New York, 1976. MR 0512360 (58:23621)
  • [3] A. Athavale, Holomorphic kernels and commuting operators, Trans. Amer. Math. Soc. 304 (1987), 101-110. MR 906808 (88m:47039)
  • [4] E. Bishop, A generalization of the Stone-Weierstrass Theorem, Pacific J. Math. 11 (1961), 777-783. MR 0133676 (24:A3502)
  • [5] N. Bourbaki, Éléments de mathématique, Livre VI, Intégration, Chapter 6, Intégration vectorielle, Hermann, Paris, 1959.
  • [6] J. Chaumat, Adherence faible étoile d'algèbres de fractions rationelles, Publ. Math. Orsay 147 (1975). MR 0442692 (56:1073)
  • [7] J. B. Conway, Subnormal operators, Pitman, London, 1981. MR 634507 (83i:47030)
  • [8] -, A course in Functional analysis, Springer-Verlag, New York, 1985. MR 768926 (86h:46001)
  • [9] -, The minimal normal extension of a function of a subnormal operator, Analysis at Urbana II, Proc. Special Year in Modern Analysis at Univ. of Illinois, 1986-87, Cambridge Univ. Press, Cambridge, 1989, pp. 128-140. MR 1009188 (91h:47021)
  • [10] -, Towards a functional calculus for subnormal tuples: The minimal normal extension and approximation in several complex variables, Proc. Sympos. Pure Math., vol. 51, Part 2, Amer. Math. Soc., Providence, R. I., 1990, pp. 105-112. MR 1077381
  • [11] J. B. Conway and R. F. Olin, A functional calculus for subnormal operators. II, Mem. Amer. Math. Soc. No. 184(1977). MR 0435913 (55:8864)
  • [12] R. E. Curto, Applications of several complex variables to multiparameter spectral theory, Surveys of Some Recent Results in Operator Theory, vol. 2, Longman, London, 1988. MR 976843 (90d:47007)
  • [13] -, Spectral inclusion for doubly commuting subnormal $ n$-tuples, Proc. Amer. Math. Soc. 83 (1981), 730-734. MR 630045 (82j:47030)
  • [14] J. Dudziak, Spectral mapping theorems for subnormal operations, J. Funct. Anal. 56 (1984), 360-387. MR 743847 (85j:47024)
  • [15] -, The minimal normal extension problem for subnormal operators, J. Funct. Anal. 65 (1986), 314-338. MR 826430 (87m:47057)
  • [16] -, On a function-theoretic weak-star density problem connected with subnormal operators, Proc. Amer. Math. Soc. 107 (1989), 679-686. MR 1017226 (90k:30072)
  • [17] T. W. Gamelin, Uniform algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969. MR 0410387 (53:14137)
  • [18] T. W. Gamelin, P. Russo, and J. E. Thomson, A Stone-Weierstrass Theorem for weak-star approximation by rational functions, J. Funct. Annl. 87 (1989), 170-176. MR 1025885 (91h:46093)
  • [19] I. Glicksberg, Measures orthogonal to algebras and sets of antisymmetry, Trans. Amer. Math. Soc. 105 (1962), 415-435. MR 0173957 (30:4164)
  • [20] T. Ito, On the commutative family of subnormal operators, J. Fac. Sci. Hokkaido Univ. 14 (1958), 1-15. MR 0107177 (21:5902)
  • [21] S. G. Krantz, Function theory of several complex variables, Wiley, New York, 1982. MR 635928 (84c:32001)
  • [22] A. Lubin, Weighted shifts and products of subnormal operators, Indiana Univ. Math. J. 26 (1977), 838-845. MR 0448139 (56:6448)
  • [23] F. Morgan, Geometric measure theory, Academic Press, Boston, Mass., 1988. MR 933756 (89f:49036)
  • [24] R. F. Olin, Functional relationships between a subnormal operator and its minimal normal extension, Pacific J. Math. 63 (1976), 221-229. MR 0420324 (54:8338)
  • [25] M. Putinar, Spectral inclusion for subnormal $ n$-tuples, Proc. Amer. Math. Soc. 90 (1984), 405-406. MR 728357 (85f:47029)
  • [26] W. Rudin, Function theory in polydiscs, Benjamin, New York, 1969. MR 0255841 (41:501)
  • [27] -, Function theory in the ball of $ {\mathbb{C}^n}$, Springer-Verlag, New York, 1980.
  • [28] R. Saerens, Interpolation theory in $ {\mathbb{C}^n}$: A survey, Complex Analysis (S. G. Krantz, Ed.), Lecture Notes in Math., vol. 1268, Springer-Verlag, Berlin, 1987. MR 907059 (88i:32021)
  • [29] D. Sarason, Weak-star density of polynomials, J. Reine Angew. Math. 252 (1972), 1-15. MR 0295088 (45:4156)
  • [30] J. L. Taylor, The analytic functional calculus for several commuting operators, Acta Math. 125 (1970), 1-30. MR 0271741 (42:6622)
  • [31] K. Yan, Invariant subspaces for joint subnormal systems (preprint).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B20, 32E30, 47A20, 47A60

Retrieve articles in all journals with MSC: 47B20, 32E30, 47A20, 47A60


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1005077-X
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society