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Free holomorphic functions and interpolation

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Abstract

In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna–Pick interpolation problem for analytic functions with positive real parts on the open unit disc. Given a function \(f : \Lambda \to \mathbb {C}\) , where \(\Lambda\) is an arbitrary subset of the open unit ball \(\mathbb{B}_n:=\{z\in \mathbb {C}^n: \|z\| < 1\}\) , we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball \([B({\mathcal H})^n]_1\) such that

$${\rm Re} \ g \geq 0 \quad {\rm and} \quad g(z)=f(z),\quad z\in \Lambda,$$

where \(B({\mathcal H})\) is the algebra of all bounded linear operators on a Hilbert space \({\mathcal H}\) . The proof employs several results from noncommutative multivariable operator theory and a noncommutative Cayley transform (introduced and studied in the present paper) acting from the set of all free holomorphic functions with positive real parts to the set of all bounded free holomorphic functions. All the results of this paper are obtained in the more general setting of free holomorphic functions with operator-valued coefficients. As consequences, we deduce some results concerning operator-valued analytic interpolation on the unit ball \({\mathbb B}_n\).

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References

  1. Agler J. and McCarthy J.E. (2002). Pick Interpolation and Hilbert Function Spaces. American Mathematical Society, Providence

    MATH  Google Scholar 

  2. Arias A. and Popescu G. (2000). Noncommutative interpolation and Poisson transforms. Israel J. Math. 115: 205–234

    Article  MATH  MathSciNet  Google Scholar 

  3. Arveson W.B. (2000). The curvature invariant of a Hilbert module over \({\mathbb C} [z_1,\ldots, z_n]\) J. Reine Angew. Math. 522: 173–236

    MATH  MathSciNet  Google Scholar 

  4. Ball, J.A., Trent, T.T., Vinnikov, V.: Interpolation and Commutant Lifting for Multipliers on Reproducing Kernels Hilbert Spaces, Operator Theory and Analysis: The M.A. Kaashoek Anniversary Volume, pp. 89–138, OT 122, Birkhauser-Verlag, Basel (2001)

  5. Ball, J.A., Vinnikov, V.: Lax–Phillips scattering and conservative linear systems: a Cuntz-algebra multidimensional setting. Mem. Am. Math. Soc. 837, iv + 101 pp (2005)

    Google Scholar 

  6. Cuntz J. (1977). Simple C*-algebras generated by isometries. Commun. Math. Phys. 57: 173–185

    Article  MATH  MathSciNet  Google Scholar 

  7. Davidson K.R., Katsoulis E. and Pitts D.R. (2001). The structure of free semigroup algebras. J. Reine Angew. Math. 533: 99–125

    MATH  MathSciNet  Google Scholar 

  8. Davidson K.R. and Pitts D.R. (1998). Nevanlinna–Pick interpolation for noncommutative analytic Toeplitzalgebras. Integr. Equat. Oper. Theory 31: 321–337

    Article  MATH  MathSciNet  Google Scholar 

  9. Davidson K.R. and Pitts D.R. (1998). The algebraic structure of non-commutative analytic Toeplitz algebras. Math. Ann. 311: 275–303

    Article  MATH  MathSciNet  Google Scholar 

  10. Davidson K.R. and Pitts D.R. (1999). Invariant subspaces and hyper-reflexivity for free semigroup algebras. Proc. Lond. Math. Soc. 78: 401–430

    Article  MathSciNet  Google Scholar 

  11. Eschmeier J. and Putinar M. (2002). Spherical contractions and interpolation problems on the unit ball. J. Reine Angew. Math. 542: 219–236

    MATH  MathSciNet  Google Scholar 

  12. Foias C. and Frazho A.E. (1990). The Commutant Lifting Approach to Interpolation Problems, Operator Theory: Advances and Applications. Birhäuser Verlag, Bassel

    Google Scholar 

  13. Helton J.W. (2002). “Positive” noncommutative polynomials are sums of squares. Ann. Math. 156(2): 675–694

    Article  MATH  MathSciNet  Google Scholar 

  14. Helton J.W. and McCullough S.A. (2004). A Positivstellensatz for non-commutative polynomials. Trans. Am. Math. Soc. 356(9): 3721–3737

    Article  MATH  MathSciNet  Google Scholar 

  15. Helton J.W., McCullough S.A. and Putinar M. (2004). A non-commutative Positivstellensatz on isometries. J. Reine Angew. Math. 568: 71–80

    MATH  MathSciNet  Google Scholar 

  16. Helton J.W., McCullough S.A. and Vinnikov V. (2006). Noncommutative convexity arises from linear matrix inequalities. J. Funct. Anal. 240(1): 105–191

    Article  MATH  MathSciNet  Google Scholar 

  17. Herglotz G. (1911). Über Potenzreien mit positiven, reelen Teil im Einheitkreis, Berichte über dieVerhaundlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig. Math. Phys. Klasse 63: 501–511

    Google Scholar 

  18. McCarthy J.E. and Putinar M. (2005). Positivity aspects of the Fantappiè transform. J. Anal. Math. 97: 57–82

    Article  MathSciNet  Google Scholar 

  19. Muhly P.S. and Solel B. (1998). Tensor algebras over C*-correspondences: representations, dilations and C*-envelopes. J. Funct. Anal. 158: 389–457

    Article  MATH  MathSciNet  Google Scholar 

  20. Muhly P.S. and Solel B. (2004). Hardy algebras, W*-correspondences and interpolation theory. Math. Ann. 330: 353–415

    Article  MATH  MathSciNet  Google Scholar 

  21. Nevanlinna R. (1919). Über beschränkte Functionen, die in gegebenen Punkten vorgeschribene Werteannehmen. Ann. Acad. Sci. Fenn. Ser A 13(1), 71

    Google Scholar 

  22. Paulsen V.I., Popescu G. and Singh D. (2002). On Bohr’s inequality. Proc. Lond. Math. Soc. 85: 493–512

    Article  MATH  MathSciNet  Google Scholar 

  23. Pick G. (1916). Über die Beschränkungen analytischer Functionen welche durch vorgebene Functionswerte bewirkt sind. Math. Ann. 77: 7–23

    Article  Google Scholar 

  24. Popescu G. (1989). Isometric dilations for infinite sequences of noncommuting operators. Trans. Am. Math. Soc. 316: 523–536

    Article  MATH  Google Scholar 

  25. Popescu G. (1989). Characteristic functions for infinite sequences of noncommuting operators. J. Oper. Theory 22: 51–71

    MATH  Google Scholar 

  26. Popescu G. (1989). Multi-analytic operators and some factorization theorems. Indiana Univ. Math. J. 38: 693–710

    Article  MATH  MathSciNet  Google Scholar 

  27. Popescu G. (1991). Von Neumann inequality for (B(H)n)1. Math. Scand. 68: 292–304

    MATH  MathSciNet  Google Scholar 

  28. Popescu G. (1992). On intertwining dilations for sequences of noncommuting operators. J. Math. Anal. Appl. 167: 382–402

    Article  MATH  MathSciNet  Google Scholar 

  29. Popescu G. (1995). Functional calculus for noncommuting operators. Mich. Math. J. 42: 345–356

    Article  MATH  Google Scholar 

  30. Popescu G. (1995). Multi-analytic operators on Fock spaces. Math. Ann. 303: 31–46

    Article  MATH  MathSciNet  Google Scholar 

  31. Popescu G. (1996). Noncommutative disc algebras and their representations. Proc. Am. Math. Soc. 124: 2137–2148

    Article  MATH  Google Scholar 

  32. Popescu G. (1998). Interpolation problems in several variables. J. Math. Anal. Appl. 227: 227–250

    Article  MATH  MathSciNet  Google Scholar 

  33. Popescu G. (1999). Poisson transforms on some C*-algebras generated by isometries. J. Funct. Anal. 161: 27–61

    Article  MATH  MathSciNet  Google Scholar 

  34. Popescu G. (2001). Curvature invariant for Hilbert modules over free semigroup algebras. Adv. Math. 158: 264–309

    Article  MATH  MathSciNet  Google Scholar 

  35. Popescu G. (2003). Similarity and ergodic theory of positive linear maps. J. Reine Angew. Math. 561: 87–129

    MATH  MathSciNet  Google Scholar 

  36. Popescu G. (2003). Multivariable Nehari problem and interpolation. J. Funct. Anal. 200: 536–581

    Article  MATH  MathSciNet  Google Scholar 

  37. Popescu, G.: Entropy and multivariable interpolation. Mem. Am. Math. Soc. 184(868), vi + 83 pp (2006)

    Google Scholar 

  38. Popescu G. (2006). Operator theory on noncommutative varieties. Indiana Univ. Math. J. 55(2): 389–442

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  40. Popescu, G.: Unitary invariants in multivariable operator theory. Mem. Am. Math. Soc. (2008) (in press)

  41. Popescu, G.: Noncommutative transforms and free pluriharmonic functions. Preprint (2007)

  42. Riesz F. (1911). Sur certains systèmes singuliers d’équations intégrales. Ann. Sci. Ecole Norm. Sup. (Paris) 28: 33–62

    MathSciNet  Google Scholar 

  43. Rudin W. (1980).Function Theory in the Unit Ball of \({\mathbb C} ^n\). Springer, New York

    MATH  Google Scholar 

  44. Sarason D. (1967). Generalized interpolation in H ∞. Trans. Am. Math. Soc. 127: 179–203

    Article  MATH  MathSciNet  Google Scholar 

  45. Stinespring W.F. (1955). Positive functions on C*-algebras. Proc. Am. Math. Soc. 6: 211–216

    Article  MATH  MathSciNet  Google Scholar 

  46. Sz.-Nagy B. and Korány K. (1957). Relations d’un problème de Nevanlinna et Pick avec la théorie des opérateurs de l’espace Hilbertien. Acta Math. Acad. Sci. Hung. 7: 295–302

    Article  Google Scholar 

  47. Sz.-Nagy, B., Foiaş, C.: Harmonic Analysis of Operators on Hilbert Space. North Holland, New York (1970)

  48. von Neumann J. (1929). Allgemeine Eigenwerttheorie Hermitescher Functionaloperatoren. Math. Ann. 102: 49–131

    Article  MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX, 78249, USA

    Gelu Popescu

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  1. Gelu Popescu
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Correspondence to Gelu Popescu.

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Research supported in part by an NSF grant.

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Popescu, G. Free holomorphic functions and interpolation. Math. Ann. 342, 1–30 (2008). https://doi.org/10.1007/s00208-008-0219-2

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  • DOI: https://doi.org/10.1007/s00208-008-0219-2

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