Errata for:
Composition Operators on Spaces of Analytic Functions

Carl C. Cowen, Purdue University
Barbara D. MacCluer, University of Richmond and University of Virginia

Errata

While we hoped to have an error free book, it did not happen!

• Introduction
  • page 7, line -3, Exercise 1.1.9: should be "j \geq 0" instead of "j>0"
• Analysis Background
  • page 23, line -5: sentence should read "the Bergman space ... with \beta(s)^2=(s+1)^{-1}" instead of "\beta(s)"
  • page 28, Exercise 2.1.13(b): sentence should read "the Bergman space ... choice \beta(s)^2=(s+1)^{-1}" instead of "\beta(s)"
  • page 30, first display: ")" missing in first integral
  • page 55: In Corollary 2.47, replace "{ z : r < |z| < 1 }" by "{ z : |\zeta - z| < 1-r }" in the statement and the proof (2 places)
    (this makes the set \Gamma(\zeta,\alpha) \cap { z : |\zeta - z| < 1-r } convex)
  • page 79: Corollary to Theorem 2.64 should be added Patch
  • page 93, Exercise 2.4.1: "\phi(\sigma(z))" should be "\sigma(\phi(z))"
  • page 93, Exercise 2.4.7: As Example 2.58 indicates, one of the hypotheses is omitted; the exercise should read " ...with \phi'(1)=1 and \phi is in the halfplane/translation case, then \phi maps ..."
• Norms
  • page 127, Exercise 3.1.7: upper bound should be should be " \left( ( 1 + |\phi(0)| )/( 1 - |\phi(0)| ) \right)^{1/p} "
  • page 127, Exercise 3.1.9: missing hypothesis: " | \lambda | = 1 "
  • page 129, line 16: "apply of characterization" should be "apply the characterization"
  • page 136, line -4 and line -7: "\|\C\|_e" should be "\|\C\|_e^2"
  • page 139, line 7: "N_{\phi}(w)" should be "N_{\phi}(a)"
  • page 166, line 2 of proof of Lemma 3.40: "\zeta=n=e_1" should be "\zeta=\eta=e_1"
• Small Spaces
  • page 176, third display of Thm. 4.1: "boundary B_N" should be "boundary D"
  • page 179, line -6: "radius 4" should be "radius 2"
  • page 195, last line: "operator" should be "operators"
• Large Spaces
  • page 202, Lemma 5.6: "|\phi(0)| = s" should be "|\phi(0)| \leq s"
  • page 203, line -6: "|\phi(0)| = s ..." should be "|\phi(0)| \leq s < ..."
    and "Y(r) = y(s,r)" should be "Y(r) \leq y(s,r)"
  • page 207, Theorem 5.11: "A > 0" should be "A \geq 1"
  • page 207, Theorem 5.11: "c independent of f" would be better said
    "c depending only on r_0, t_0, and F"
  • page 211, Notes, line -2: "no bounded, non-compact operators" should be
    "no bounded, non-compact, composition operators"
• Special Results for Several Variables
• Spectral Properties
  • page 276, line -2 of text: "[Cal62, page 331]" should be "[Cal52, page 331]" (which reference is added below)
  • page 281: Lemma 7.25 should be revised to include some non-fixed points Patch
  • page 283: Proof of Theorem 7.26 must be revised since Exercise 7.5.2 is incorrectly stated Patch
  • page 288, Theorem 7.29: Hypothesis "z_0'=z_0" was omitted.
  • page 288, Exercise 7.5.2: "\{ e^{i\theta} : |\phi_n(e^{i\theta})|=1 \}" should be
    "\{ \phi_n(e^{i\theta}) : |\phi_n(e^{i\theta})|=1 \}". This necessitates changes on pages 79, 281, 283, and 296. Patch
  • page 293, line -8: ")" should be deleted at the end of the sentence following the display
  • page 296, line 7 (preceding third display): "\phi^{(K)}" should be "\phi_K"
  • page 296, line -8: "\{ w : |w|=1 and |\phi_n(w)|=1 \}" should be
    "\{ \phi_n(w) : |w|=1 and |\phi_n(w)|=1 \}". Patch
  • page 293, line -8: ")" should be deleted at the end of the sentence
  • page 301, line 22: "halfplane/translation" should be "plane/translation"
  • page 304, line -11: "..spectrum and essential of.." should be "..spectrum and essential spectrum of.."
• Normality
  • page 311, line 8: "H" should be "H^2(\beta,D)"
  • page 312, Exercise 8.1.1(b) should read "Suppose C is a normal operator and ... "
• Miscellanea
  • page 321, first display: norm should be replaced by "(...)" in third sum
  • page 347, line -13: "interested the" should be "interested in the"
• Bibliography
  • New entry:
    [Cal52] L. Carleson, Sets of uniqueness for functions regular in the unit circle, Acta Math. 87(1952), 325-345.
  • [Hu89]: date should be "1989".

Report errors to Carl Cowen:

 ccowen@iupui.edu 

Last Update: July 30, 1996