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Ap -Inner Functions

  • Haakan Hedenmalm
  • Boris Korenblum
  • Kehe Zhu
Part of the Graduate Texts in Mathematics book series (GTM, volume 199)

Abstract

In this chapter, we introduce the notion of A p α -inner functions and prove a growth estimate for them. The A p α -inner functions are analogous to the classical inner functions which play an important role in the factorization theory of the Hardy spaces. Each A p α -inner function is extremal for a z-invariant subspace, and the ones that arise from subspaces given by finitely many zeros are called finite zero extremal functions (for α = 0, they are also called finite zero-divisors). In the unweighted case α = 0, we will prove the expansive multiplier property of A up -inner functions, and obtain an “inner-outer”-type factorization of functions in A p . In the process, we find that all singly generated invariant subspaces are generated by its extremal function. In the special case of p = 2 and α = 0, we find an analogue of the classical Carathéodory-Schur theorem: the closure of the finite zero-divisors in the topology of uniform convergence on compact subsets are the A2-subinner functions. In particular, all A2-inner functions are norm approximable by finite zero-divisors.

Keywords

Hardy Space Invariant Subspace Extremal Problem Bergman Space Blaschke Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Haakan Hedenmalm
    • 1
  • Boris Korenblum
    • 2
  • Kehe Zhu
    • 2
  1. 1.Department of MathematicsLund UniversityLundSweden
  2. 2.Department of MathematicsState University of New York at AlbanyAlbanyUSA

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