Mathematische Annalen

, Volume 370, Issue 1–2, pp 423–446 | Cite as

A non-commutative Julia inequality

  • John E. McCarthyEmail author
  • James E. Pascoe


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.


  1. 1.
    Abate, M.: The Julia–Wolff–Carathéodory theorem in polydisks. J. Anal. Math. 74, 275–306 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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. 3.
    Agler, J., McCarthy, J.E.: Global holomorphic functions in several non-commuting variables. Can. J. Math. 67(2), 241–285 (2015)CrossRefzbMATHGoogle Scholar
  4. 4.
    Agler, J., McCarthy, J.E.: Operator theory and the Oka extension theorem. Hiroshima Math. J. 45(1), 9–34 (2015)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Agler, J., McCarthy, J.E.: Pick interpolation for free holomorphic functions. Am. J. Math. 137(6), 1685–1701 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Agler, J., McCarthy, J.E.: Aspects of non-commutative function theory. Concr. Oper. 3, 15–24 (2016)MathSciNetzbMATHGoogle Scholar
  7. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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)Google Scholar
  12. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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. 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)Google Scholar
  15. 15.
    Bickel, K., Knese, G.: Inner functions on the bidisk and associated Hilbert spaces. J. Funct. Anal. 265(11), 2753–2790 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Carathéodory, C.: Über die Winkelderivierten von beschränkten analytischen Funktionen. Sitzunber. Preuss. Akad. Wiss. 4, 39–52 (1929)Google Scholar
  17. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Helton, J.W., Klep, I., McCullough, S.: Proper analytic free maps. J. Funct. Anal. 260(5), 1476–1490 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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)zbMATHGoogle Scholar
  21. 21.
    Jafari, F.: Angular derivatives in polydisks. Indian J. Math. 35, 197–212 (1993)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Julia, G.: Extension nouvelle d’un lemme de Schwarz. Acta Math. 42, 349–355 (1920)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jury, M.T.: An Improved Julia–Caratheodory Theorem for Schur–Agler Mappings of the Unit Ball. arXiv:0707.3423
  24. 24.
    Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Foundations of Free Non-commutative Function Theory. AMS, Providence (2014)zbMATHGoogle Scholar
  25. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Popescu, G.: Von: Neumann inequality for \((B({\cal{H}})^{n})_{1}\). Math. Scand. 68, 292–304 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Popescu, G.: Free holomorphic functions on the unit ball of \(B({\cal{H}})^n\). J. Funct. Anal. 241(1), 268–333 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Popescu, G.: Free biholomorphic classification of noncommutative domains. Int. Math. Res. Not. IMRN 4, 784–850 (2011)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Rudin, W.: Zeros of holomorphic functions in balls. Indag. Math. 38, 57–65 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Sarason, D.: Sub-Hardy Hilbert Spaces in the Unit Disk. University of Arkansas Lecture Notes. Wiley, New York (1994)Google Scholar
  34. 34.
    Taylor, J.L.: A general framework for a multi-operator functional calculus. Adv. Math. 9, 183–252 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Taylor, J.L.: Functions of several non-commuting variables. Bull. Am. Math. Soc. 79, 1–34 (1973)CrossRefGoogle Scholar
  36. 36.
    von Neumann, J.: Eine, Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr. 4, 258–281 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wlodarczyk, K.: Julia’s lemma and Wolff’s theorem for \(J*\)-algebras. Proc. Am. Math. Soc. 99, 472–476 (1987)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Washington University in Saint LouisSt. LouisUSA

Personalised recommendations