Composition Operators on Spaces of Analytic Functions

Barbara D. MacCluer, University of Richmond and University of Virginia

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".

- New entry:

cowen@math.purdue.edu

Last Update: July 30, 1996