Complex Analysis and Operator Theory

, Volume 12, Issue 4, pp 945–967 | Cite as

\(\hbox {H}^2\) Spaces of Non-commutative Functions



We define the Hardy spaces of free noncommutative functions on the noncommutative polydisc and the noncommutative ball and study their basic properties. Our technique combines the general methods of noncommutative function theory and asymptotic formulae for integration over the unitary group. The results are the first step in developing the general theory of free noncommutative bounded symmetric domains on the one hand and in studying the asymptotic free noncommutative analogues of classical spaces of analytic functions on the other.


Non-commutative functions Hardy \(\hbox {H}^2\) spaces Asymptotic free independence Non-commutative polydisc Non-commutative ball Taylor–Taylor expansion for non-commutative functions 


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© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at San AntonioSan AntonioUSA
  2. 2.Institute of Mathematics ‘Simion Stoilow’ of the Romanian AcademyBucharestRomania
  3. 3.Department of MathematicsBen Gurion University of NegevBe’er ShevaIsrael

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