Operator Analysis: Hilbert Space Methods In Complex Analysis

Contents on Cambridge University Press Website

Corrections

We are very grateful to Orr Shalit for carefully reading the book, and catching most of the following mistakes.

P. 30, Line 17: The reference should be to von Neumann’s 1929 paper, [209].
P. 62, Line 9-: ker( $\mathbb D$ ) should be Her( $\mathbb D$ ).

P. 65, Equation 2.91: We need to know that $\sup \| f(w) \|$ is finite. This requires first proving that $f$ is locally bounded. This can be done along the lines of W. Rudin’s proof of the one variable result in Thm. 3.31 of his book “Functional Analysis”, where in d variables one uses the Cauchy formula on the distinguished boundary of a polydisk instead of the circle in his argument.

P. 72, Lemma 3.2: The proof shows that $\frac{1 - \phi(\lambda) \phi(\mu)^*}{1 - \overline{\mu} \lambda}$ is positive semi-definite. To get the assertion of the Lemma, one then applies this to the function $\phi(\overline{\lambda})^*$ .

P. 82, Line 4: $u$ is a rational map from $\mathbb D$ into $\mathcal{ B(H,M)}$ .
P. 101, Line 9: The third term on the left-hand side should be $L(\lambda - \lambda_0) \gamma$ .
P. 101, Line 7-: The = should be $\geq$ .
P. 103, Line 6: $\mathcal{W_{B(X)}}$ should be just $\mathcal{W}$ .
P. 106, Line 1-: orthogonal projections should be orthogonal complements.
P. 114, Lines 7- to 1-: three of the $\mathbb{D}^2$ should be $\mathbb{D}^2 \times \mathbb{D}^2$ .
P. 149, Line 7: $h$ should be $f$ in Eq. 6.6.
P. 171, Line 5: $\mu$ should be $\bar \mu$ .
P. 191, Line 3-: $\theta_T$ should be $\sin(\theta_T)$ .
P. 213, Line 2: Delete “with”.
P. 217, Line 16: There should be no square on $\Vert \phi \Vert_{{\mathbb D}^2}$ .

P. 233, Line 5-: The B. and F. Delyon family should be the set of $T$ that satisfy $\overline{W(T)} \subset C$ , since the spectrum of an operator is contained in the closure of its numerical range, but not necessarily in the numerical range itself (for example the backward shift has numerical range the open unit disk).

P. 243: “subordinate to $\mathbb{D}$ ” should be “subordinate to ${\mathbb D}^d$ .”
P. 263: Line 12: It should be (10.27) that is called the primal; (10.28), using the matrix $A$ from (10.27), is the dual problem.

P. 270: Line 18: $\Delta$ is in $SA{\mathbb M}_n^d$ .
P. 273: Line 13: Should be “that $0 \leq t \leq \varepsilon$ ,”.
P. 288: Line 7-: ${ d \choose 2}$ should be ${n \choose 2}$ .
P. 299: Line 8: “their ampliations” should be “its ampliations”.
P. 326, Line 9: add ” and $B_\delta \subseteq \Omega$ .”
P. 332, Line 14: ${\mathcal M}^d$ should be ${\mathbb M}^d$ .
P. 337, Line 13: Indeed, the boundary of $\Phi( {\rm Ball}(a^{(k)}, r))$ is disjoint from ${\rm Ball}(b^{(k)},s)$ , so the latter must be in its interior.

P. 340, Line 3: Missing comma.
P. 346, Line2: “domain” should be “function”.
P. 350, Line 8: “dense” should be “sequentially dense”.
P. 354, Line 7-: This is bad notation. $A$ should be $\alpha$
throughout the proof. Unless $\delta(0) = 0$ , $\alpha$ will in
general be distinct from $A$ in (16.30).

Last updated: November 11, 2023

John McCarthy, Ph.D.

Operator Analysis: Hilbert Space Methods In Complex Analysis