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)

 
 

 

The minimal normal extension for $ M\sb z$ on the Hardy space of a planar region


Author: John Spraker
Journal: Trans. Amer. Math. Soc. 318 (1990), 57-67
MSC: Primary 47B20; Secondary 47B15, 47B38
DOI: https://doi.org/10.1090/S0002-9947-1990-1008703-3
MathSciNet review: 1008703
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Multiplication by the independent variable on $ {H^2}(R)$ for $ R$ a bounded open region in the complex plane $ \mathbb{C}$ is a subnormal operator. This paper characterizes its minimal normal extension $ N$. Any normal operator is determined by a scalar-valued spectral measure and a multiplicity function. It is a consequence of some standard operator theory that a scalar-valued spectral measure for $ N$ is harmonic measure for $ R$, $ \omega $. This paper investigates the multiplicity function $ m$ for $ N$. It is shown that $ m$ is bounded above by two $ \omega $-a.e., and necessary and sufficient conditions are given for $ m$ to attain this upper bound on a set of positive harmonic measure. Examples are given which indicate the relationship between $ N$ and the boundary of $ R$.


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

  • [1] M. B. Abrahamse, Multiplication operators,, Lecture Notes in Math., vol. 693, Springer-Verlag, Berlin, 1978, pp. 17-36. MR 526530 (80b:47042)
  • [2] M. B. Abrahamse and R. G. Douglas, A class of subnormal operators related to multiply-connected domains, Adv. in Math. 19 (1976), 106-148. MR 0397468 (53:1327)
  • [3] M. B. Abrahamse and T. Kriete, The spectral multiplicity of a multiplication operator, Indiana Univ. Math. J. 22 (1973), 845-857. MR 0320797 (47:9331)
  • [4] L. Ahlfors, Conformal invariants, topics in geometric function theory, McGraw-Hill, New York, 1973. MR 0357743 (50:10211)
  • [5] M. Brelot, On topologies and boundaries in potential theory, Lecture Notes in Math., vol. 175, Springer-Verlag, Berlin, 1971. MR 0281940 (43:7654)
  • [6] C. Caratheodory, Uber die Begrenzung einfach Zusammenhangerder Gebiete, Math. Ann. 73 (1913), 323-370. MR 1511737
  • [7] E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Univ. Press, New York, 1966. MR 0231999 (38:325)
  • [8] J. B. Conway, A course in functional analysis, Springer-Verlag, New York, 1985. MR 768926 (86h:46001)
  • [9] -, Functions of one complex variable (2nd ed.), Springer-Verlag, New York, 1986.
  • [10] -, The minimal normal extension of a function of a subnormal operator, Preprint, 1987.
  • [11] -, Subnormal operators, Pitman, 1981.
  • [12] J. Dudziak, The minimal normal extension problem for subnormal operators, J. Funct. Anal. 65 (1986), 314-338. MR 826430 (87m:47057)
  • [13] P. L. Duren, Theory of $ {H^p}$ space, Academic Press, New York, 1970.
  • [14] S. D. Fisher, Function theory on planar domains, Wiley, New York, 1983. MR 694693 (85d:30001)
  • [15] T. W. Gamelin, Uniform algebras, (2nd ed.), Chelsea, New York, 1984.
  • [16] M. Hasumi, Hardy classes on infinitely connected Riemann surfaces, Lecture Notes in Math., vol. 1027, Springer-Verlag, Berlin, 1983. MR 723502 (85k:30066)
  • [17] L. L. Helms, Introduction to potential theory, Wiley-Interscience, New York, 1969. MR 0261018 (41:5638)
  • [18] T. L. Kriete, An elementary approach to the multiplicity theory of multiplication operators, Rocky Mountain J. Math. 16 (1986), 23-33. MR 829193 (87e:47035)
  • [19] K. Kuratowski, Topology, Vol. 1, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [20] A. J. Lohwater and W. Seidel, An example in conformal mapping, Duke Math J. 15 (1948), 137-143. MR 0023903 (9:420f)
  • [21] A. I. Markushevitch, Theory of functions of a complex variable, (2nd ed.), Chelsea, New York, 1985.
  • [22] R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941), 137-172. MR 0003919 (2:292h)
  • [23] J. McMillan and G. Piranian, Compression and expansion of boundary sets, Duke Math. J. 40 (1973), 599-605. MR 0318492 (47:7039)
  • [24] R. L. Moore, Concerning triods in the plane and the junction points of plane continua, Proc. Nat. Acad. Sci. U.S.A. 14 (1928), 85-88. MR 0000642 (1:107a)
  • [25] M. A. Naimark, Normed rings, Noordhoff, Groningen, 1964, pp. 515-519. MR 0355601 (50:8075)
  • [26] M. H. A. Newman, Elements of the topology of plane sets of points, (2nd ed.), Cambridge Univ. Press, New York, 1954. MR 0044820 (13:483a)
  • [27] M. Ohtsuka, Dirichlet problem, extremal length and prime ends, Van Nostrand Reinhold, New York, 1970.
  • [28] A. Pfluger, Lectures on conformal mapping, Dept. of Math., Indiana University, 1969.
  • [29] C. Pommerenke, Univalent functions, Vanderhoeck & Ruprecht, Gottingen, 1975. MR 0507768 (58:22526)
  • [30] F. Riesz and M. Riesz, Über die Randwerte einer analytischen Funktion $ 4$, Congr. Scand. Math., Stockholm, 1916, pp. 27-44.
  • [31] W. Rudin, Analytic functions of class $ {H^p}$, Trans. Amer. Math. Soc. 78 (1955), 46-66. MR 0067993 (16:810b)
  • [32] M. Tsuji, Potential theory in modern function theory, (2nd ed.), Chelsea, New York, 1975. MR 0414898 (54:2990)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B20, 47B15, 47B38

Retrieve articles in all journals with MSC: 47B20, 47B15, 47B38


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-1008703-3
Keywords: Harmonic measure, multiplication operator, minimal normal extension
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society