From 343843f281bc7cd2a17d2ea49c347e36eafe405f Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sun, 18 Oct 2026 21:49:15 -0400 Subject: [PATCH] Updated draft. --- src/op/system/numerical-radius.tex | 10 +++++----- src/op/system/schur.tex | 8 ++++---- 2 files changed, 9 insertions(+), 9 deletions(-) diff --git a/src/op/system/numerical-radius.tex b/src/op/system/numerical-radius.tex index 05d9218..48909ad 100644 --- a/src/op/system/numerical-radius.tex +++ b/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*} diff --git a/src/op/system/schur.tex b/src/op/system/schur.tex index 6528e9e..2f746ad 100644 --- a/src/op/system/schur.tex +++ b/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} \\