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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2145))

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Abstract

This chapter will present some basic facts from complex analysis, operator theory and von Neumann algebras. These results will be needed in the sequel.

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Bibliography

  1. W. Arveson, An Invitation to C*-Algebras. Graduate Texts in Mathematics, vol. 39 (Springer, New York, 1998)

    Google Scholar 

  2. M. Armstrong, Basic Topology. Undergraduate Texts in Mathematics (Springer, New York, 1983) [Corrected reprint of the 1979 original]

    Google Scholar 

  3. A. Besicovitch, On sufficient condition for a function to be analytic and on behavior of analytic functions in the neighborhood of non-isolated singular points. Proc. Lond. Math. Soc. 32, 1–9 (1931)

    Article  MathSciNet  Google Scholar 

  4. C. Berenstein, R. Gay, Complex Variables: An Introduction. Graduate Texts in Mathematics, vol. 125 (Springer, New York, 1991)

    Google Scholar 

  5. B. Blackadar, Operator Algebras. Encyclopaedia of Mathematical Sciences, vol. 122 (Springer, Berlin, 2006) [Theory of C ∗-algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, III]

    Google Scholar 

  6. L. Carleson, T. Gamelin, Complex Dynamics (Springer, Berlin, 1993)

    Book  MATH  Google Scholar 

  7. X. Chen, K. Guo, Analytic Hilbert Modules. π-Chapman & Hall/CRC Research Notes in Mathematics, vol. 433, (2003)

    Google Scholar 

  8. E. Collingwood, A. Lohwater, Theory of Cluster Sets (Cambridge University Press, Cambridge, 1966)

    Book  MATH  Google Scholar 

  9. J. Conway, A Course in Operator Theory. Graduate Studies in Mathematics, vol. 21 (American Mathematical Society, Rhode Island, 2000)

    Google Scholar 

  10. C. Cowen, The commutant of an analytic Toeplitz operator. Trans. Am. Math. Soc. 239, 1–31 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  11. K. Davidson, C*-Algebras by Example. Fields Institute Monographs, vol. 6 (American Mathematical Society, Rhode Island, 1996)

    Google Scholar 

  12. J. Dixmier, von Neumann Algebras (North-Holland, Amsterdam, 1981)

    MATH  Google Scholar 

  13. R. Douglas, M. Putinar, K. Wang, Reducing subspaces for analytic multipliers of the Bergman space. J. Funct. Anal. 263, 1744–1765 (2012) arXiv: math.FA/ 1110.4920v1

    Google Scholar 

  14. P. Duren, A. Schuster, Bergman Spaces. Mathematics Surveys and Monographs, vol. 100 (American Mathematical Society, Rhode Island, 2004)

    Google Scholar 

  15. R. Douglas, S. Sun, D. Zheng, Multiplication operators on the Bergman space via analytic continuation. Adv. Math. 226, 541–583 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Dudziak, Vitushkin’ Conjecture for Removable Sets. Universitext (Springer, New York, 2010)

    Book  Google Scholar 

  17. O. Frostman, Potentiel d’equilibre et capacite des ensembles avec quelques applications \(\grave{a}\) la theorie des fonctions. Medd. Lunds Mat. Sem. 3, 1–118 (1935)

    Google Scholar 

  18. J. Garnett, Bounded Analytic Functions (Academic, New York, 1981)

    MATH  Google Scholar 

  19. K. Guo, H. Huang, On multiplication operators of the Bergman space: similarity, unitary equivalence and reducing subspaces. J. Oper. Theory 65, 355–378 (2011)

    MATH  MathSciNet  Google Scholar 

  20. P. Gorkin, R. Mortini, Value distribution of interpolating Blaschke product. J. Lond. Math. Soc. 72, 151–168 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. P. Gorkin, R. Mortini, Radial limits of interpolating Blaschke product. Math. Ann. 331, 417–444 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. G. Goluzin, Geometric Theory of Functions of a Complex Variable, vol. 26 (American Mathematical Society, 1969), p. 255

    Google Scholar 

  23. K. Guo, Operator theory and Von Neumann algebras (a manuscript)

    Google Scholar 

  24. K. Guo, S. Sun, D. Zheng, C. Zhong, Multiplication operators on the Bergman space via the Hardy space of the bidisk. J. Reine Angew. Math. 629, 129–168 (2009)

    MathSciNet  Google Scholar 

  25. P. Halmos, Measure Theory (Van Nostrand, Princeton, 1950)

    Book  MATH  Google Scholar 

  26. W. Hastings, A Carleson measure theorem for Bergman spaces. Proc. Am. Math. Soc. 52, 237–241 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  27. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces (Springer, New York, 2000)

    Book  MATH  Google Scholar 

  28. K. Hoffman, Banach Spaces of Analytic Functions (Prentice-Hall, Englewood Cliffs, 1962)

    MATH  Google Scholar 

  29. K. Hoffman, Bounded analytic functions and Gleason parts. Ann. Math. 86, 74–111 (1967)

    Article  MATH  Google Scholar 

  30. C. Horowitz, Facorization theorems for functions in the Bergman spaces. Duke Math. J. 44, 201–213 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  31. L. Hormander, Introduction to Complex Analysis in Several Variables, 3rd edn (North-Holland, Amsterdam, 1990)

    Google Scholar 

  32. H. Huang, Maximal abelian von Neumann algebras and Toeplitz operators with separately radial symbols. Integr. Equ. Oper. Theory 64, 381–398 (2009)

    Article  MATH  Google Scholar 

  33. H. Huang, von Neumann algebras generated by multiplication operators on the weighted Bergman space: a function-theory view into operator theory. Sci. China Ser. A 56, 811–822 (2013)

    Google Scholar 

  34. V. Jones, von Neumann Algebras, UC Berkeley Mathematics (2009). http://math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf

  35. J. Kelley, General Topology. Graduate Texts in Mathematics, vol. 27 (Springer, New York, 1955)

    Google Scholar 

  36. S. Krantz, Function Theory of Several Complex Variables. Pure and Applied Mathematics (A Wiley-Interscience Publication, New York, 1982)

    Google Scholar 

  37. K. Lawson, Some lemmas on interpolating Blaschke products and a correction. Can. J. Math. 21, 531–534 (1969)

    Article  MATH  Google Scholar 

  38. D. Luecking, A technique for characterizing Carleson measures on Bergman spaces. Proc. Am. Math. Soc. 87, 656–660 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  39. D. Marshall, Removable sets for bounded analytic functions. J. Math. Sci. 26, 2232–2234 (1984)

    Article  Google Scholar 

  40. J. Mashreghi, Derivatives of Inner Functions. The Fields Institute for Research in the Mathematical Sciences. Fields Institute Monographs, vol. 31 (Springer, New York, 2013)

    Google Scholar 

  41. G. McDonald, C. Sundberg, Toeplitz operators on the disc. Indiana Univ. Math. J. 28, 595–611 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  42. J. Milnor, Dynamics in One Complex Variable. Annals of Mathematics Studies, vol. 160 (Princeton University Press, Princeton, 2006)

    Google Scholar 

  43. Z. Nehari, Conformal Mapping (McGraw-Hill, New York, 1952)

    MATH  Google Scholar 

  44. R. Nevanlinna, Eindeutige analytische Funktionen, 2te Aufl. (Springer, Berlin, 1953)

    Book  Google Scholar 

  45. K. Oka, Sur les fonctions de plusieurs variables II. Domaines d’holomorphie. J. Sci. Hiroshima Univ. 7, 115–130 (1937)

    Google Scholar 

  46. W. Rudin, Function Theory in Polydiscs (Benjamin, New York, 1969)

    MATH  Google Scholar 

  47. W. Rudin, Function Theory in the UnIt ball of \(\mathbb{C}^{n}\). Grundlehren der Mathematischen, vol. 241 (Springer, New York, 1980)

    Google Scholar 

  48. W. Rudin, Real and Complex Analysis, 3rd edn. (McGraw-Hill Book Co., New York, 1987)

    MATH  Google Scholar 

  49. W. Rudin, A generalization of a theorem of Frostman. Math. Scand. 21, 136–143 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  50. K. Seip, Beuling type density theorems in the unit disk. Invent. Math. 113, 21–39 (1994)

    Article  MathSciNet  Google Scholar 

  51. H. Shapiro, Comparative Approximation in Two Topologies. Approximation Theory, vol. 4 (Banach Center Publication, Warsaw, 1979), 225–232

    Google Scholar 

  52. S. Sun, D. Zheng, C. Zhong, Classification of reducing subspaces of a class of multiplication operators via the Hardy space of the bidisk. Can. J. Math. 62, 415–438 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  53. S. Sun, D. Zheng, C. Zhong, Multiplication operators on the Bergman space and weighted shifts. J. Oper. Theory 59, 435–452 (2008)

    MATH  MathSciNet  Google Scholar 

  54. J. Thomson, The commutant of a class of analytic Toeplitz operators. Am. J. Math. 99, 522–529 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  55. W. Veech, A Second Course in Complex Analysis (Benjamin, New York, 1967)

    MATH  Google Scholar 

  56. A. Weil, L’integrable de Cauchy et les fonctions de plusieurs variables. Math. Ann. 111, 178–182 (1935)

    Article  MathSciNet  Google Scholar 

  57. K. Zhu, Reducing subspaces for a class of multiplication operators. J. Lond. Math. Soc. 62, 553–568 (2000)

    Article  MATH  Google Scholar 

  58. K. Zhu, Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics, vol. 226 (Springer, New York, 2005)

    Google Scholar 

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Guo, K., Huang, H. (2015). Some Preliminaries. In: Multiplication Operators on the Bergman Space. Lecture Notes in Mathematics, vol 2145. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46845-6_2

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