Skip to main content

Advertisement

SpringerLink
Log in

A non-commutative Julia inequality

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We prove a Julia inequality for bounded non-commutative functions on polynomial polyhedra. We use this to deduce a Julia inequality for holomorphicfunctions on classical domains in \(\mathbb {C}^d\). We look at differentiability at a boundary point for functions that have a certain regularity there.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. It is more common to consider the row contractions, but we choose column contractions so that what we call the distinguished boundary will be non-empty. It is easy to pass between these two sets, since the column contractions are just the adjoints of the row contractions.

References

  1. Abate, M.: The Julia–Wolff–Carathéodory theorem in polydisks. J. Anal. Math. 74, 275–306 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agler, J., McCarthy, J.E.: Global holomorphic functions in several non-commuting variables II. Can. Math. Bull. (2017). doi:10.4153/CMB-2017-044-4

  3. Agler, J., McCarthy, J.E.: Global holomorphic functions in several non-commuting variables. Can. J. Math. 67(2), 241–285 (2015)

    Article  MATH  Google Scholar 

  4. Agler, J., McCarthy, J.E.: Operator theory and the Oka extension theorem. Hiroshima Math. J. 45(1), 9–34 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Agler, J., McCarthy, J.E.: Pick interpolation for free holomorphic functions. Am. J. Math. 137(6), 1685–1701 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Agler, J., McCarthy, J.E.: Aspects of non-commutative function theory. Concr. Oper. 3, 15–24 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Agler, J., McCarthy, J.E., Young, N.J.: Facial behavior of analytic functions on the bidisk. Bull. Lond. Math. Soc. 43, 478–494 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Agler, J., McCarthy, J.E., Young, N.J., Carathéodory, A.: theorem for the bidisk via Hilbert space methods. Math. Ann. 352(3), 581–624 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Agler, J., McCarthy, J.E., Young, N.J.: On the representation of holomorphic functions on polyhedra. Mich. Math. J. 62(4), 675–689 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Agler, J., Tully-Doyle, R., Young, N.J.: Boundary behavior of analytic functions of two variables via generalized models. Indag. Math. (N.S.) 23(4), 995–1027 (2012). doi:10.1016/j.indag.2012.07.003

    Article  MathSciNet  MATH  Google Scholar 

  11. Alpay, D., Kalyuzhnyi-Verbovetzkii, D.S.: Matrix-\(J\)-unitary non-commutative rational formal power series. In: The State Space Method Generalizations and Applications, vol. 161, pp. 49-113 . Birkhäuser, Basel (2006)

  12. Balasubramanian, S.: Toeplitz corona and the Douglas property for free functions 0022–247X. J. Math. Anal. Appl. 428(1), 1–11 (2015). doi:10.1016/j.jmaa.2015.03.005

    Article  MathSciNet  MATH  Google Scholar 

  13. Ball, J.A., Marx, G., Vinnikov, V.: Interpolation and Transfer Function Realization for the Non-commutative Schur–Agler Class (To appear). arXiv:1602.00762

  14. Ball, J.A., Groenewald, G., Malakorn, T.: Conservative structured noncommutative multidimensional linear systems. In: The State Space Method Generalizations and Applications, vol. 161, pp. 179–223. Birkhäuser, Basel (2006)

  15. Bickel, K., Knese, G.: Inner functions on the bidisk and associated Hilbert spaces. J. Funct. Anal. 265(11), 2753–2790 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Carathéodory, C.: Über die Winkelderivierten von beschränkten analytischen Funktionen. Sitzunber. Preuss. Akad. Wiss. 4, 39–52 (1929)

  17. Cimpric, J., Helton, J.W., McCullough, S., Nelson, C.: A noncommutative real nullstellensatz corresponds to a noncommutative real ideal: algorithms. Proc. Lond. Math. Soc. (3) 106(5), 1060–1086 (2013). doi:10.1112/plms/pds060

    Article  MathSciNet  MATH  Google Scholar 

  18. Helton, J.W., Klep, I., McCullough, S.: Proper analytic free maps. J. Funct. Anal. 260(5), 1476–1490 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Helton, J.W., McCullough, S.: Every convex free basic semi-algebraic set has an LMI representation. Ann. Math. (2) 176(2), 979–1013 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hervé, M.: Quelques propriétés des applications analytiques d’une boule à\(m\) dimensions dan elle-même. J. Math. Pures Appl. 9(42), 117–147 (1963)

    MATH  Google Scholar 

  21. Jafari, F.: Angular derivatives in polydisks. Indian J. Math. 35, 197–212 (1993)

    MathSciNet  MATH  Google Scholar 

  22. Julia, G.: Extension nouvelle d’un lemme de Schwarz. Acta Math. 42, 349–355 (1920)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jury, M.T.: An Improved Julia–Caratheodory Theorem for Schur–Agler Mappings of the Unit Ball. arXiv:0707.3423

  24. Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Foundations of Free Non-commutative Function Theory. AMS, Providence (2014)

    MATH  Google Scholar 

  25. McCarthy, J.E., Timoney, R.: Non-commutative automorphisms of bounded non-commutative domains. Proc. R. Soc. Edinburgh Sect. A 146(5), 1037–1045 (2016). doi:10.1017/S0308210515000748

    Article  MathSciNet  MATH  Google Scholar 

  26. Muhly, P.S., Solel, B.: Tensorial function theory: from Berezin transforms to Taylor’s Taylor series and back. Integral Equ. Oper. Theory 76(4), 463–508 (2013). doi:10.1007/s00020-013-2062-4

    Article  MathSciNet  MATH  Google Scholar 

  27. Pascoe, J.E.: The inverse function theorem and the Jacobian conjecture for free analysis. Math. Z. 278(3–4), 987–994 (2014). doi:10.1007/s00209-014-1342-2

    Article  MathSciNet  MATH  Google Scholar 

  28. Pascoe, J.E., Tully-Doyle, R.: Free pick functions: representations, asymptotic behavior and matrix monotonicity in several noncommuting variables. J. Funct. Anal. 273(1), 283–328 (2017). doi:10.1016/j.jfa.2017.04.001

    Article  MathSciNet  MATH  Google Scholar 

  29. Popescu, G.: Von: Neumann inequality for \((B({\cal{H}})^{n})_{1}\). Math. Scand. 68, 292–304 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. Popescu, G.: Free holomorphic functions on the unit ball of \(B({\cal{H}})^n\). J. Funct. Anal. 241(1), 268–333 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Popescu, G.: Free biholomorphic classification of noncommutative domains. Int. Math. Res. Not. IMRN 4, 784–850 (2011)

    MathSciNet  MATH  Google Scholar 

  32. Rudin, W.: Zeros of holomorphic functions in balls. Indag. Math. 38, 57–65 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sarason, D.: Sub-Hardy Hilbert Spaces in the Unit Disk. University of Arkansas Lecture Notes. Wiley, New York (1994)

    Google Scholar 

  34. Taylor, J.L.: A general framework for a multi-operator functional calculus. Adv. Math. 9, 183–252 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  35. Taylor, J.L.: Functions of several non-commuting variables. Bull. Am. Math. Soc. 79, 1–34 (1973)

    Article  Google Scholar 

  36. von Neumann, J.: Eine, Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr. 4, 258–281 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wlodarczyk, K.: Julia’s lemma and Wolff’s theorem for \(J*\)-algebras. Proc. Am. Math. Soc. 99, 472–476 (1987)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Washington University in Saint Louis, St. Louis, MO, USA

    John E. McCarthy & James E. Pascoe

Authors
  1. John E. McCarthy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James E. Pascoe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John E. McCarthy.

Additional information

Communicated by Andreas Thom.

J. E. McCarthy partially supported by National Science Foundation Grant DMS 1565243. J. E. Pascoe partially supported by National Science Foundation Fellowship DMS 1606260.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

McCarthy, J.E., Pascoe, J.E. A non-commutative Julia inequality. Math. Ann. 370, 423–446 (2018). https://doi.org/10.1007/s00208-017-1596-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-017-1596-1

Search

Navigation