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Non-commutative functional calculus

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Abstract

We develop a functional calculus for d-tuples of non-commuting elements in a Banach algebra. The functions we apply are free analytic functions, that is, nc-functions that are bounded on certain polynomial polyhedra.

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References

  1. J. Agler and J. E. McCarthy, Pick interpolation for free holomorphic functions, Amer. J. Math. 137 (2015), 1685–1701.

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Alpay and D. S. Kalyuzhnyi-Verbovetzkii, Matrix-J-unitary non-commutative rational formal power series, The State Space Method, Generalizations and Applications, Birkhäuser Verlag Basel/Switzerland, 2006, pp. 49–113.

    Book  Google Scholar 

  5. J. A. Ball, G. Groenewald, and T. Malakorn, Conservative structured noncommutative multidimensional linear systems, The state space method general-izations and applications, Birkhäuser Verlag Basel/Switzerland, 2006, pp. 179–223.

    Book  MATH  Google Scholar 

  6. R. E. Curto, Applications of several complex variables to multiparameter spectral theory, Surveys of Some Recent Results in Operator Theory, Longman Scientific and Technical, 1988, pp. 25–90.

    MATH  Google Scholar 

  7. J. Diestel, J. H. Fourie, and J. Swart, The Metric Theory of Tensor Products, American Mathematical Society, Providence, RI, 2008.

    Book  MATH  Google Scholar 

  8. J. W. Helton, I. Klep, and S. McCullough, Analytic mappings between noncommutative pencil balls, J. Math. Anal. Appl. 376 (2011), 407–428.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov, Foundations of Free Noncommutative Function Theory, Amer. Math. Soc., Providence, RI, 2014.

    Book  MATH  Google Scholar 

  12. V. I. Paulsen, Completely bounded homomorphisms of operator algebras, Proc. Amer.Math. Soc. 92 (1984), 225–228.

    Article  MathSciNet  MATH  Google Scholar 

  13. V. I. Paulsen, Every completely polynomially bounded operator is similar to a contraction, J. Funct. Anal. 55 (1984), 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Pisier, Introduction to Operator Space Theory, Cambridge University Press, Cambridge, 2003.

    Book  MATH  Google Scholar 

  15. G. Popescu, Free holomorphic functions on the unit ball of B(H)n, J. Funct. Anal. 241 (2006), 268–333.

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Popescu, Free holomorphic functions and interpolation, Math. Ann. 342 (2008), no. 1, 1–30.

  17. G. Popescu, Free holomorphic automorphisms of the unit ball of B(H)n, J. Reine Angew. Math. 638 (2010), 119–168.

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  19. G. Popescu, Free biholomorphic functions and operator model theory, J. Funct. Anal. 262 (2012), 3240–3308.

    Article  MathSciNet  MATH  Google Scholar 

  20. G. Popescu, Free biholomorphic functions and operator model theory, II, J. Funct. Anal. 265 (2013), 786–836.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Putinar, Uniqueness of Taylor’s functional calculus, Proc. Amer. Math. Soc. 89 (1983), 647–650.

    MathSciNet  MATH  Google Scholar 

  22. G-C. Rota, On models for linear operators, Comm. Pure Appl. Math. 13 (1960), 469–472.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. A. Ryan, Introduction to Tensor Products of Banach spaces, Springer-Verlag London, Ltd., London, 2002.

    Book  MATH  Google Scholar 

  24. B. Szokefalvi-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert Space, second edition, Springer, New York, 2010.

    Book  Google Scholar 

  25. J. L. Taylor, The analytic functional calculus for several commuting operators, Acta Math. 125 (1970), 1–38.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6 (1970), 172–191.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. L. Taylor, A general framework for a multi-operator functional calculus, Advances in Math. 9 (1972), 183–252.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. UCSD, La Jolla, CA, 92093, USA

    Jim Agler

  2. Washington University, St. Louis, MO, 63130, USA

    John E. McCarthy

Authors
  1. Jim Agler
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  2. John E. McCarthy
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Corresponding author

Correspondence to John E. McCarthy.

Additional information

Partially supported by National Science Foundation Grant DMS 1361720.

Partially supported by National Science Foundation Grant DMS 1300280.

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Agler, J., McCarthy, J.E. Non-commutative functional calculus. JAMA 137, 211–229 (2019). https://doi.org/10.1007/s11854-018-0070-7

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  • DOI: https://doi.org/10.1007/s11854-018-0070-7

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