## John McCarthy, Ph.D.

Operator Theory, One and Several Complex Variables, and Their Interaction

# Pick Interpolation and Hilbert Function Spaces

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. 67, Proposition 5.38
George Tsikalas points out that the hypotheses of Proposition 5.38 should be the same as Proposition 5.37, with the assumption that the space be holomorphic
replaced by the assumption that $\| M_z \| = 1$. The proof given does not use the three point Pick property, but does use that Schwarz’s lemma is satisfied.

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. 144, Question 9.57.
The answer is yes. See “Interpolating sequences in spaces with the complete Pick property,” A. Aleman, M. Hartz, J. McCarthy and Stefan Richter, InternationalMathematical Research Notices, [2019] (12) 3832-3854  PDF

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: September 30, 2020