The minimal normal extension for 𝑀_{𝑧} on the Hardy space of a planar region

Spraker, John

doi:10.1090/S0002-9947-1990-1008703-3

The minimal normal extension for $M_ z$ on the Hardy space of a planar region
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Trans. Amer. Math. Soc. 318 (1990), 57-67 Request permission

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

M. B. Abrahamse, Multiplication operators, Hilbert space operators (Proc. Conf., Calif. State Univ., Long Beach, Calif., 1977) Lecture Notes in Math., vol. 693, Springer, Berlin, 1978, pp. 17–36. MR 526530
M. B. Abrahamse and R. G. Douglas, A class of subnormal operators related to multiply-connected domains, Advances in Math. 19 (1976), no. 1, 106–148. MR 397468, DOI 10.1016/0001-8708(76)90023-2
M. B. Abrahamse and Thomas L. Kriete, The spectral multiplicity of a multiplication operator, Indiana Univ. Math. J. 22 (1972/73), 845–857. MR 320797, DOI 10.1512/iumj.1973.22.22072
Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
Marcel Brelot, On topologies and boundaries in potential theory, Lecture Notes in Mathematics, Vol. 175, Springer-Verlag, Berlin-New York, 1971. Enlarged edition of a course of lectures delivered in 1966. MR 0281940, DOI 10.1007/BFb0060353
C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete, Math. Ann. 73 (1913), no. 3, 323–370 (German). MR 1511737, DOI 10.1007/BF01456699
E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 0231999, DOI 10.1017/CBO9780511566134
John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR 768926, DOI 10.1007/978-1-4757-3828-5

Functions of one complex variable

The minimal normal extension of a function of a subnormal operator

Subnormal operators

James Dudziak, The minimal normal extension problem for subnormal operators, J. Funct. Anal. 65 (1986), no. 3, 314–338. MR 826430, DOI 10.1016/0022-1236(86)90022-4

Theory of

space

Stephen D. Fisher, Function theory on planar domains, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A second course in complex analysis; A Wiley-Interscience Publication. MR 694693

Uniform algebras

Morisuke Hasumi, Hardy classes on infinitely connected Riemann surfaces, Lecture Notes in Mathematics, vol. 1027, Springer-Verlag, Berlin, 1983. MR 723502, DOI 10.1007/BFb0071447
L. L. Helms, Introduction to potential theory, Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1969. MR 0261018
Thomas L. Kriete III, An elementary approach to the multiplicity theory of multiplication operators, Rocky Mountain J. Math. 16 (1986), no. 1, 23–32. MR 829193, DOI 10.1216/RMJ-1986-16-1-23
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
A. J. Lohwater and W. Seidel, An example in conformal mapping, Duke Math. J. 15 (1948), 137–143. MR 23903, DOI 10.1215/S0012-7094-48-01516-6

Theory of functions of a complex variable

Robert S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941), 137–172. MR 3919, DOI 10.1090/S0002-9947-1941-0003919-6
John E. McMillan and George Piranian, Compression and expansion of boundary sets, Duke Math. J. 40 (1973), 599–605. MR 318492
R. L. Moore, Concerning the open subsets of a plane continuum, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 24–25. MR 642, DOI 10.1073/pnas.26.1.24
M. A. Naĭmark, Normed rings, Reprinting of the revised English edition, Wolters-Noordhoff Publishing, Groningen, 1970. Translated from the first Russian edition by Leo F. Boron. MR 0355601
M. H. A. Newman, Elements of the topology of plane sets of points, Cambridge, at the University Press, 1951. 2nd ed. MR 0044820

Dirichlet problem, extremal length and prime ends

Lectures on conformal mapping

Christian Pommerenke, Univalent functions, Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen. MR 0507768

Über die Randwerte einer analytischen Funktion

Walter Rudin, Analytic functions of class $H_p$, Trans. Amer. Math. Soc. 78 (1955), 46–66. MR 67993, DOI 10.1090/S0002-9947-1955-0067993-3
M. Tsuji, Potential theory in modern function theory, Chelsea Publishing Co., New York, 1975. Reprinting of the 1959 original. MR 0414898