Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

Angular Derivatives at Boundary Fixed Points for Self-Maps of the Disk

Pietro Poggi-Corradini
Proceedings of the American Mathematical Society
Vol. 126, No. 6 (Jun., 1998), pp. 1697-1708
Stable URL: http://www.jstor.org/stable/118575
Page Count: 12
  • Read Online (Free)
  • Download ($30.00)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Angular Derivatives at Boundary Fixed Points for Self-Maps of the Disk
Preview not available

Abstract

Let φ be a one-to-one analytic function of the unit disk D into itself, with φ (0) = 0. The origin is an attracting fixed point for φ , if φ is not a rotation. In addition, there can be fixed points on ∂ D where φ has a finite angular derivative. These boundary fixed points must be repelling (abbreviated b.r.f.p.). The Koenigs function of φ is a one-to-one analytic function σ defined on D such that φ =σ -1(λ σ), where λ =φ ′(0). If φ K is the first iterate of φ that does have b.r.f.p., we compute the Hardy number of σ , h(σ ) = sup{p > 0: σ ∈ Hp( D)}, in terms of the smallest angular derivative of φ K at its b.r.f.p.. In the case when no iterate of φ has b.r.f.p., then $\sigma \in \bigcap _{p<\infty}$Hp, and vice versa. This has applications to composition operators, since σ is a formal eigenfunction of the operator Cφ(f) = f ⚬ φ . When Cφ acts on H2( D), by a result of C. Cowen and B. MacCluer, the spectrum of Cφ is determined by λ and the essential spectral radius of Cφ, re(Cφ). Also, by a result of P. Bourdon and J. Shapiro, and our earlier work, re(Cφ) can be computed in terms of h(σ ). Hence, our result implies that the spectrum of Cφ is determined by the derivative of φ at the fixed point 0 ∈ D and the angular derivatives at b.r.f.p. of φ or some iterate of φ .

Page Thumbnails

  • Thumbnail: Page 
1697
    1697
  • Thumbnail: Page 
1698
    1698
  • Thumbnail: Page 
1699
    1699
  • Thumbnail: Page 
1700
    1700
  • Thumbnail: Page 
1701
    1701
  • Thumbnail: Page 
1702
    1702
  • Thumbnail: Page 
1703
    1703
  • Thumbnail: Page 
1704
    1704
  • Thumbnail: Page 
1705
    1705
  • Thumbnail: Page 
1706
    1706
  • Thumbnail: Page 
1707
    1707
  • Thumbnail: Page 
1708
    1708