Pick Interpolation and Hilbert Function Spaces

By Jim Agler and John E. McCarthy

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Corrections

P. 29, Proof 2 of Theorem 2.53. This second proof is wrong. The sets $Y_\alpha$ in the middle of page 30 are not closed (unless the Hilbert space is finite dimensional). The other applications of Kurosh's theorem throughout the book are correct. The point is that if you want to fill in an infinite matrix indexed on $X \times X$,, and can do it on every finite subset of $X$ in a way that hangs together, then you can do it on all of $X$ at once. But you should wait until the end to realize this matrix as a Grammian - otherwise you have too many choices, and its hard to keep things under control.

P. 43, Proof of Theorem 3.21. The last sentence should read: "Moreover, by Theorem 3.7, we have $$ r^n \hat{F}(n) = \int f(re^{i\theta}) e^{-i n \theta} d\sigma(\theta). $$

P. 52, Example 4.9.

Stefan Maurer points out that for $s > 0$,
$(n+1)^{-s}$ are the moments
of the measure $1/(\Gamma(s)) (log(1/x))^(s-1) dx$ on the interval
$[0,1]$, so the spaces ${\cal H}_s$ are also $P^2(\mu)$ spaces for
a radial measure $\mu$.
Therefore their multiplier algebras are isometrically $H^\infty
(\Bbb D)$.

P. 81, Theorem 7.6.

Delete the top line on p. 83.
The proof as given requires $k$ to be positive definite.
Gorazd Bizjak in his thesis "Generalizations of the Pick problem"
proves that if $F_N$ is positive semi-definite, then
$k$ is necessarily positive definite.

P. 90, Corollary 7.41.

The spaces $P^2(\mu_s)$ are not defined for $s \leq 0$, so should be
eliminated from the statement of the Corollary.

P. 92, Lemma 7.47.

The assumption that the diagonal entries of A be positive
is unnecessary. See Theorem 3.3 in "Matrices: Theory and
Applications" by Dennis Serre, Springer, 2002.

P. 94, Question 7.55.

Antonio Serra in "New Examples of non-complete Pick kernels",
IEOT 53 [2005] No. 4, pp. 553-572, constructs a family of kernels that
have the Pick property but not the complete Pick property.

P. 115, Remark 8.66.

The theorem of Greene, Richter and Sundberg does not require that $\mathcal
L$ be one-dimensional: it is valid for any

$\mathcal L$.

P. 123, Proposition 8.83.

The functions $f_t(\lambda)$ should be defined to be
$f_t(\lambda) = \lambda^1 \sqrt{ 1 + t (\lambda^2)^2}$,
where $(\lambda^2)^2$ means the square of the second coordinate of
$\lambda$.

P. 124, Line 1. "any $w_3$ in some neighborhood of
$\frac{1}{2\sqrt{2}}$ is allowable".

P. 129, Definition 9.10.

After "closed unit ball" insert "of the multiplier algebra".

P. 134, Proof of Theorem 9.19 (ii).

This is a little confusing. See
9.19 erratum for a better exposition.

P. 140, Theorem 9.43.

This theorem is due to H. Shapiro and A. Shields [SS62]
for the Dirichlet space, and for more general Pick kernels
it was proved by D. Marshall and C. Sundberg [MS94].

P. 143, Exercise 9.54.

Line 6-: Define c_i = \frac{ \bar \lambda_i}{\lambda_i} \prod_{j \neq i} \frac{ \bar \lambda_j}{\lambda_j} \frac{\lambda_j - \lambda_i}{1 - \bar \lambda_j \lambda_i}

On lines 1- and 3- change one of the "i"'s in the numerator to
a "j".

Line 1-: Replace c_i \bar c_j with its reciprocal.

P. 146, Line 15:

Multiply c_i in the denominator by \frac{\bar \lambda_i}{\lambda_i}

P. 147, Lines 2, 10,11,12,13,15:

Multiply the expression by ( \frac{ \bar \lambda_i \lambda_j }{ | \lambda_i \lambda_j |} )^2

P. 162, Theorem 10.29.

In the statement of the theorem in the case i=2 (the dilation
version of the commutant lifting theorem), it should be added that
$\cal H$ is semi-invariant for $Y_2$, and condition (i) should
become
$P_{\cal H} Y_2 |_{\cal H} = X$.

P. 169, Line 8- and P. 175, Line 4

It should read "Replacing $L(T)$ by $\frac{1}{2} ( L(T) +
\overline{L(T*)})$".

P. 193, Question 11.92.

The answer to the question is "yes", provided the problem
is minimal (an $N$-point extremal problem is minimal if none of the
$N-1$ point subproblems is extremal). See "Toral Algebraic Sets and
Function Theory on Polydisks" by J. Agler, J. McCarthy and M.
Stankus (J. Geom. Anal., to appear).
The minimality condition can be removed with a little extra work.

P. 211, Definition 13.4.

A third condition must be added to the set of test functions:
For any finite set $\lambda_1, \dots, \lambda_N$, the set of
matrices representable as $\{ \sum_{\beta}
[ 1 - \langle b_\beta (\lambda_j), b_\beta(\lambda_i) \rangle ]
\Gamma_\beta(\lambda_i,\lamda_j) : \Gamma_\beta \geq 0,
\beta \in {\cal I} \}$ must be closed.

P. 212, Theorem 13.7.

An extra hypothesis is needed.
For every finite set $F \subseteq X$, the set of matrices
$\Gamma_\beta$ for which the representation in (ii) holds on $F$
must be compact in the direct product over ${\cal I}$ of all
$card(F) \times card(F)$ matrices.

*Last updated: June 22, 2006*