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
\nu(f_n)

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
\psi_{\alpha,0,b}(\pi)

(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
"Maschioni".

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:
math.msu.edu
in the file:
pub/papers/shapiro

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

... Joel Shapiro  (shapiro@math.msu.edu)