Updated draft.

src/op/system/numerical-radius.tex

@@ -51,7 +51,7 @@
 
     so $|\dpn{T\xi, \xi}{H}| \le 1$, and $w(T) \le 1$. 
     
-    (3): For any $\xi \in H$ with $\norm{\xi}_H = 1$, $|\dpn{T\xi, \xi}{H}| \le \norm{T}_{B(H)}$, so $w(T) \le \norm{T}_H$. On the other hand, if $w(T) \le 1$, then for each $\xi, \eta \in H$ with $\dpn{T\xi, \eta}{H} \in \real$, 
+    (3): For any $\xi \in H$ with $\norm{\xi}_H = 1$, $|\dpn{T\xi, \xi}{H}| \le \norm{T}_{B(H)}$, so $w(T) \le \norm{T}_{B(H)}$. On the other hand, if $w(T) \le 1$, then for each $\xi, \eta \in H$ with $\dpn{T\xi, \eta}{H} \in \real$, 
     \begin{align*}
         \dpn{T\xi, \eta}{H} &= \frac{1}{4}\sum_{k = 0}^3 i^k\dpn{T(\xi + i^k\eta), \xi + i^k\eta}{H} \\
         &= \frac{1}{4}(\dpn{T(\xi + \eta), \xi + \eta}{H} - \dpn{T(\xi - \eta), \xi - \eta}{H}) \\
@@ -65,7 +65,7 @@
     \norm{T\xi}_H^2 &\le \norm{\xi}_H^2
     \end{align*}
 
-    (4): For any $\xi \in H$ with $\norm{\xi}_H = 1$, $|\dpn{T\xi, \xi}{H}| \le \norm{T}_{B(H)}$, so $w(T) \le \norm{T}_H$. The sharp case is given by $I_H$. 
+    (4): For any $\xi \in H$ with $\norm{\xi}_H = 1$, $|\dpn{T\xi, \xi}{H}| \le \norm{T}_{B(H)}$, so $w(T) \le \norm{T}_{B(H)}$. The sharp case is given by $I_H$. 
 
     On the other hand, if $w(T) \le 1$, then $2 + T + T^* = 2 + 2\text{Re}(T) \ge 0$ and $2 - T - T^* = 2 - 2\text{Re}(T) \ge 0$ imply that $\norm{\text{Re}(T)}_{B(H)} \le 1$. Similarly, $\norm{\text{Im}(T)}_{B(H)} \le 1$ as well, so $\norm{T}_{B(H)} \le \norm{\text{Re}(T)}_{B(H)} + \norm{\text{Im}(T)}_{B(H)} \le 2$. Therefore $\norm{T}_{B(H)} \le 2w(T)$. 
 
@@ -97,7 +97,7 @@
 
 \begin{proposition}
 \label{proposition:cyclic-forward-shift-radius}
-    Let $H$ be a complex Hilbert space, $T \in B(H)$, $R = \sum_{j = 2}^nE_{(j+1)j} \otimes T \in M_n(B(H))$, $\bracsn{e_j}_0^{n-1} \subset \complex^n$ be the canonical basis for $\complex^n$, and
+    Let $H$ be a complex Hilbert space, $T \in B(H)$, $R = \sum_{j = 1}^{n-1}E_{(j+1)j} \otimes T \in M_n(B(H))$, $\bracsn{e_j}_0^{n-1} \subset \complex^n$ be the canonical basis for $\complex^n$, and
     \[
     S_n: \complex^n \to \complex^n \quad S_ne_j = e_{j+1 \mod n}
     \]
@@ -127,7 +127,7 @@
     (2): Let $\bracsn{f_j}_{0}^{n-1} \subset \complex^n$ be an orthonormal eigenbasis for $S_n$, with eigenvalues $\bracsn{\omega^k}_0^{n-1}$. For any $\bracsn{\xi_j}_0^{n-1} \subset H$, let $\xi = \sum_{j = 0}^{n-1}\xi_j \otimes f_j$, then 
     \begin{align*}
         \dpn{(T \otimes S_n)\xi, \xi}{H^n} &= \sum_{j, k = 0}^{n-1}\dpn{(T \otimes S_n)(\xi_j \otimes f_j), \xi_k \otimes f_k}{H^n} \\
-        &= \sum_{j, k = 0}^{n-1}\dpn{T\xi_j, \xi_k}{H} \dpn{S_nf_j, f_k}{H} = \sum_{j = 0}^{n-1}\omega^j\dpn{T\xi_j, \xi_j}{H} \\
+        &= \sum_{j, k = 0}^{n-1}\dpn{T\xi_j, \xi_k}{H} \dpn{S_nf_j, f_k}{\complex^n} = \sum_{j = 0}^{n-1}\omega^j\dpn{T\xi_j, \xi_j}{H} \\
         |\dpn{(T \otimes S_n)\xi, \xi}{H^n}| &\le \sum_{j = 0}^{n-1}|\omega^j| \cdot w(T) \cdot \norm{\xi_j}_{H}^2 = w(T)\norm{\xi}_{H^n}^2
     \end{align*}
 
@@ -140,7 +140,7 @@
 
     (3): Let $\xi = (\xi_1, \cdots, \xi_{n}) \in H^n$, then 
     \begin{align*}
-        \dpn{R\xi, \xi}{H^n} &= \sum_{j = 2}^{n}\dpn{(E_{(j+1)j}\otimes T)\xi, \xi}{H^n} = \sum_{j = 1}^{n-1}  \dpn{T\xi_j, \xi_{j+1}}{H} \\
+        \dpn{R\xi, \xi}{H^n} &= \sum_{j = 1}^{n-1}\dpn{(E_{(j+1)j}\otimes T)\xi, \xi}{H^n} = \sum_{j = 1}^{n-1}  \dpn{T\xi_j, \xi_{j+1}}{H} \\
         &= \dpn{T0, \xi_1}{H} + \dpn{T\xi_n, 0}{H} + \sum_{j = 1}^{n-1}  \dpn{T\xi_j, \xi_{j+1}}{H} \\
         &= \dpn{(T \otimes S_{n+1})(\xi_1, \cdots, \xi_n, 0), (\xi_1, \cdots, \xi_n, 0)}{H^{n+1}}
     \end{align*}

src/op/system/schur.tex

@@ -7,15 +7,15 @@
     Let $H$ be a complex Hilbert space, $A = \sum_{i, j = 1}^n E_{ij} \otimes a_{ij} \in M_n(B(H))$ be positive, and $\Lambda = \sum_{i, j = 1}^n \lambda_{ij}E_{ij} \in M_n(\complex)$ be positive, then $S_A(\Lambda) = \sum_{i, j = 1}^n \lambda_{ij}E_{ij} \otimes a_{ij}$ is also positive. 
 \end{lemma}
 \begin{proof}
-    For any $\xi \in \complex^n$, the projection map $P_\xi$ onto its span is given by $\sum_{i, j = 1}^n \xi_i E_{ij} \ol{\xi_j}$. In which case, 
+    For any unit vector $\xi \in \complex^n$, the projection map $P_\xi$ onto its span is given by $\sum_{i, j = 1}^n \xi_i E_{ij} \ol{\xi_j}$. In which case, 
     \begin{align*}
         S_A(P_\xi) &= \sum_{i, j = 1}^n \xi_i \ol{\xi_j} E_{ij} \otimes a_{ij} \\
-        &= \braks{\sum_{i = 1}^n \xi_{i}E_{ii}}\braks{\sum_{i, j = 1}^n \xi_i \ol{\xi_j} E_{ij} \otimes a_{ij}}\braks{\sum_{j = 1}^n {\xi_j}E_{jj}}^* \ge 0
+        &= \braks{\sum_{i = 1}^n \xi_{i}E_{ii}}\braks{\sum_{i, j = 1}^n E_{ij} \otimes a_{ij}}\braks{\sum_{j = 1}^n {\xi_j}E_{jj}}^* \ge 0
     \end{align*}
     
 
 
-    Using the \hyperref[Spectral Theorem]{definition:linfty-functional-calculus}, let $\bracsn{\xi^{(j)}}_1^n \subset \complex^n$ be an orthonormal eigenbasis for $\Lambda$ with eigenvectors $\seqf{\lambda_j} \subset \complex$. For each $1 \le j \le n$, let $P_j \in M_n(\complex)$ be the rank one projection onto the span of $\xi^{(j)}$, then 
+    Using the \hyperref[Spectral Theorem]{definition:linfty-functional-calculus}, let $\bracsn{\xi^{(j)}}_1^n \subset \complex^n$ be an orthonormal eigenbasis for $\Lambda$ with eigenvalues $\seqf{\lambda_j} \subset \complex$. For each $1 \le j \le n$, let $P_j \in M_n(\complex)$ be the rank one projection onto the span of $\xi^{(j)}$, then 
     \[
         S_A(\Lambda) = \sum_{j = 1}^n \lambda_j S_A(P_{j}) \ge 0
     \]
@@ -45,7 +45,7 @@
 
     (2) $\Leftrightarrow$ (3): Let $q = \sum_{-N \le j \le N}\mu_j z^j$ and $\xi = (\mu_{-N}, \cdots, \mu_{N}) \in \complex^{2N+1}$, then
     \begin{align*}
-        \dpn{Pq, q}{L^2(\mathbb{T}; \complex)} &= \sum_{-N \le j, k, l \le N}\dpn{\lambda_j z^j \cdot \mu_k z^k, \mu_l z^l}{L^2(\mathbb{T}; \complex)} \\
+        \dpn{Pq, q}{L^2(\mathbb{T}; \complex)} &= \sum_{-n \le j \le n}\sum_{-N \le k, l \le N}\dpn{\lambda_j z^j \cdot \mu_k z^k, \mu_l z^l}{L^2(\mathbb{T}; \complex)} \\
         &= \sum_{-N \le k, l \le N}\dpn{\lambda_{l-k}\mu_k z^l, \mu_l z^l}{L^2(\mathbb{T}; \complex)} \\
         &= \sum_{-N \le k, l \le N} \lambda_{l - k}\mu_k\ol{\mu_l}= \sum_{k, l = 1}^{2N+1} \lambda_{l - k}\xi_{k}\ol{\xi_{l}} \\
         &= \angles{\paren{\sum_{i, j = 1}^{2N+1} E_{ij}\lambda_{i - j}}\xi, \xi} \\