Errata for:

Composition Operators and Classical Function Theory

by Joel H. Shapiro

November 4, 1994


  page 4   last line: ... neither elliptic nor hyperbolic
  page 12  equation (4): d theta, not dt

  page 12, 4 lines from bottom: ||b|| <=  ||b||_infinity
           (not the other way around).
  page 31  line 6 (just after displayed equation). 
           The limit statement should refer to: 
                f_r composed with phi,
                ^^^ ^^^^^^^^ ^^^^ ^^^
           not just to f_r. 

  page 34  Problem 7, part (a), second line: if and only if 
  page 35  four lines from bottom:  "Hunziger", not "Huniziger"

  page 40, second line of large display: f_n'(\phi(z))
           not f_n(\phi(z))  (the derivative symbol is missing)

  page 48  last line: \Psi_{\alpha,0,b}(\Pi) instead of 

                   (Capital \psi and \pi instead of lower-case)

  page 55  eight lines from the bottom: delete the second 
           hyphen in "Julia-Carathe'odory".

  page 57  last line of statement of Julia-Carath. Theorem:
           ``.. omegabar eta.." not "omega etabar"
  page 67  second line: "w_0 \in \Pi" (not "\omega_0 \in \Pi").
           four lines from bottom: "p \in \Pi" 
           (not \phi \in \Pi).                                                     

  page 71  in statement of Corollary: 
                phi is a univalent self-map  of U
           (not just a holomorphic one: the Cor. is false for        
           arbitrary self-maps, as I go to great lengths to 
           show in Chapter 10.)

  page 74  Exercise 12: 
               in second line: "boundary of U" not just "U"
               in third line:   sup_n|\phi_n'(\zeta)|  
               (subscript "n" is missing)

  page 98, Displayed statement in next-to-last paragraph:  
           "If phi is a univalent self-map ..."

  page 103, Exercise 12, part (a):
            Show that if T is compact, 
            then the diagonal entries of the matrix are the 
            only eigenvalues of T.

      (The Backward Shift on H^2 is a counterexample to the
       result of part (a) as it is stated.)

  page 110 two lines after first display: The word "define" 
           occurs twice.  Delete one! 

  page 196  Middle of page, two lines below displayed formula:
      " ... u=|f|^2 ..."  not "... u = |f'|^2 .."

  page 178  second line before the Littlewood-Paley Identity:
      "... it is more convenient to put a little more care ..."

  page 189  next-to-last line of large display:

            Instead of   ||z||^2 
            there should appear

                         ||\phi||^2 - |\phi(0)|^2

            which results from the integral of N_\phi that 
            occurs in the line above upon appplying the 
            change-of-variable formula (4) on p 179 
            to the function f(z) = z.

  page 190  right-hand side of inequality (22): 
            Integrate with respect to dA(z), not dA(w). 

  page 200  third line before last paragraph:"Mascioni", not 

  page 208  references [KrM '90] and [KrM '92]: "Kriete", not" 
            Kreite".(same error on index entry, page 218)

  page 211  reference [STa '72]  IUMJ 23 (1973)  
            (vol. 23, not vol. 125)

  page 212  reference [Sk1 '87] Composition semigroups 
                          (not  Composition operators)
In addition, I've been informed of these errors that I haven't 
checked yet:

  page 49  fourth line from top: y = Ax + Bx^\gamma

                            (not y = A + Bx^\gamma)

  page 139  In the estimates involving w_{n+1} - w_n, one 
            should not take the
            specific constants claimed there too seriously!

I'd like to thank Mihaela Poplicher, Maria Tjani, John Clifford, Aristomenis
Siskakis,  P. Galanopoulos, and Raymond Mortini for their help here.

The latest corrections are available by anonymous ftp from:
in the file:

Please let me know about any other mistakes you might encounter.

... Joel Shapiro  (