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Smooth Analytic Functions and Model Subspaces

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Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2)

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 5))

Abstract

The main themes of this survey are as follows: (a) the canonical (Riesz–Nevanlinna) factorization in various classes of analytic functions on the disk that are smooth up to its boundary, and (b) model subspaces (i.e., invariant subspaces of the backward shift) in the Hardy spaces H p and in BMOA. It is the interrelationship and a peculiar cross-fertilization between the two topics that we wish to highlight.

Dedicated to the memory of Cora Sadosky

Supported in part by grants MTM2011-27932-C02-01, MTM2014-51834-P from El Ministerio de Economía y Competitividad (Spain) and grant 2014-SGR-289 from AGAUR (Generalitat de Catalunya).

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References

  1. P.R. Ahern, D.N. Clark, On inner functions with H p-derivative. Michigan Math. J. 21, 115–127 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. R.B. Burckel, An Introduction to Classical Complex Analysis, vol. I (Academic Press, New York, 1979)

    Book  MATH  Google Scholar 

  3. C. Carathéodory, Theory of Functions of a Complex Variable, vol. II (Chelsea Publishing Company, New York, 1954)

    MATH  Google Scholar 

  4. L. Carleson, A representation formula for the Dirichlet integral. Math. Z. 73, 190–196 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  5. W.S. Cohn, Radial limits and star invariant subspaces of bounded mean oscillation. Am. J. Math. 108, 719–749 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. P.L. Duren, B.W. Romberg, A.L. Shields, Linear functionals on H p spaces with 0 < p < 1. J. Reine Angew. Math. 238, 32–60 (1969)

    Google Scholar 

  7. K.M. Dyakonov, Smooth functions and coinvariant subspaces of the shift operator. Algebra i Analiz 4 (5), 117–147 (1992); translation in St. Petersburg Math. J. 4, 933–959 (1993)

    Google Scholar 

  8. K.M. Dyakonov, Division and multiplication by inner functions and embedding theorems for star-invariant subspaces. Am. J. Math. 115, 881–902 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. K.M. Dyakonov, Multiplication by Blaschke products and stability of ideals in Lipschitz algebras. Math. Scand. 73, 246–258 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. K.M. Dyakonov, Factorization of smooth analytic functions via Hilbert–Schmidt operators. Algebra i Analiz 8 (4), 1–42 (1996); translation in St. Petersburg Math. J. 8, 543–569 (1997)

    Google Scholar 

  11. K.M. Dyakonov, Equivalent norms on Lipschitz-type spaces of holomorphic functions. Acta Math. 178, 143–167 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. K.M. Dyakonov, Two theorems on star-invariant subspaces of BMOA. Indiana Univ. Math. J. 56, 643–658 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. K.M. Dyakonov, Blaschke products and nonideal ideals in higher order Lipschitz algebras. Algebra i Analiz 21 (6), 182–201 (2009); translation in St. Petersburg Math. J. 21, 979–993 (2010)

    Google Scholar 

  14. E.M. Dyn’kin, Methods of the theory of singular integrals: Littlewood-Paley theory and its applications, in Commutative Harmonic Analysis, IV, ed. by V.P. Khavin, N.K. Nikol’skii. Encyclopaedia of Mathematical Sciences, vol. 42, pp. 97–194 (Springer, Berlin, 1992)

    Google Scholar 

  15. E.M. Dyn’kin, The pseudoanalytic extension. J. Anal. Math. 60, 45–70 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. J.B. Garnett, Bounded Analytic Functions, Revised 1st edn. (Springer, New York, 2007)

    Google Scholar 

  17. V.P. Havin, The factorization of analytic functions that are smooth up to the boundary. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 22, 202–205 (1971)

    MathSciNet  Google Scholar 

  18. V.E. Katsnel’son, Remark on canonical factorization in certain analytic function spaces. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 30, 163–164 (1972); English transl. in J. Soviet Math. 4, 444–445 (1975)

    Google Scholar 

  19. B.I. Korenblum, V.M. Faĭvyshevskiĭ, A certain class of compression operators that are connected with the divisibility of analytic functions. Ukrain. Mat. Zh. 24, 692–695 (1972); English transl. in Ukrainian Math. J. 24, 559–561 (1973)

    Google Scholar 

  20. N.K. Nikolski, Operators, Functions, and Systems: An Easy Reading, Volume 1: Hardy, Hankel, and Toeplitz. Mathematical Surveys and Monographs, vol. 92 (American Mathematical Society, Providence, 2002)

    Google Scholar 

  21. M. Pavlović, On Dyakonov’s paper “Equivalent norms on Lipschitz-type spaces of holomorphic functions”. Acta Math. 183, 141–143 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Rabindranathan, Toeplitz operators and division by inner functions. Indiana Univ. Math. J. 22, 523–529 (1972/1973)

    Google Scholar 

  23. D. Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk (Wiley, New York, 1994)

    MATH  Google Scholar 

  24. F.A. Shamoyan, Division by an inner function in certain spaces of functions that are analytic in the disc. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 22, 206–208 (1971)

    MathSciNet  MATH  Google Scholar 

  25. F.A. Shamoyan, A class of Toeplitz operators that are connected with the divisibility of analytic functions. Funktsional. Anal. i Prilozhen. 13 (1), 83 (1979); English transl. in Funct. Anal. Appl. 13, 70–71 (1979)

    Google Scholar 

  26. F.A. Shamoyan, Toeplitz operators and division by an inner function in some spaces of analytic functions. Akad. Nauk Armyan. SSR Dokl. 76 (3), 109–113 (1983, Russian)

    Google Scholar 

  27. N.A. Shirokov, Ideals and factorization in algebras of analytic functions that are smooth up to the boundary. Trudy Mat. Inst. Steklov. 130, 196–222 (1978)

    MathSciNet  MATH  Google Scholar 

  28. N.A. Shirokov, Division and multiplication by inner functions in spaces of analytic functions smooth up to the boundary, in Complex Analysis and Spectral Theory (Leningrad, 1979/1980). Lecture Notes in Mathematics, Vol. 864, pp. 413–439 (Springer, Berlin/New York, 1981)

    Google Scholar 

  29. N.A. Shirokov, Analytic Functions Smooth Up to the Boundary. Lecture Notes in Mathematics, vol. 1312 (Springer, Berlin, 1988)

    Google Scholar 

  30. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (Princeton University Press, Princeton, 1993)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Departament de Matemàtiques i Informàtica, BGSMath and Universitat de Barcelona, Gran Via 585, E-08007, Barcelona, Spain

    Konstantin M. Dyakonov

  2. ICREA, Pg. Lluís Companys 23, E-08010, Barcelona, Spain

    Konstantin M. Dyakonov

Authors
  1. Konstantin M. Dyakonov
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Correspondence to Konstantin M. Dyakonov .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico, USA

    María Cristina Pereyra

  2. Department of Mathematics, Venezuelan Institute for Scientific Research, Caracas, Venezuela

    Stefania Marcantognini

  3. Department of Mathematical Sciences, Georgia Southern University, Statesboro, Georgia, USA

    Alexander M. Stokolos

  4. Department of Mathematical and Actuarial Sciences, Roosevelt University Chicago, Chicago, Illinois, USA

    Wilfredo Urbina

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Dyakonov, K.M. (2017). Smooth Analytic Functions and Model Subspaces. In: Pereyra, M., Marcantognini, S., Stokolos, A., Urbina, W. (eds) Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2). Association for Women in Mathematics Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-51593-9_9

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