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Toni Mhax
Toni Mhax
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Taking $B=\begin{pmatrix}0&\sqrt{2}\\0&0\end{pmatrix}$ and $S=\{z\in\mathbb{C}, |z|\le \frac{1}{\sqrt{2}}\}$ is a counterexample. The bound is $\sigma_1(B)\le \sqrt{2}$.

For any matrix $B,M\in \mathbb{M}_n(\mathbb{C})$ the numerical range of $B$ is the set $W(B)=\{x^*Bx, x\in \mathbb{C}^n, x^*x=1\}$. Also $W(\dfrac{M+M^*}{2})=\mathfrak{R}(W(M))$, $W(\lambda B)=\lambda W(B)$ (rotation and homothety) and $\sigma_1(M)\le 2w(M)$ where $w(M)$ is the numerical radius of $M$, (these are known), with $\lambda B=M$. We see that if $\dfrac{M+M^*}{2}\ge -0.5I$, for any $\lambda\in S$, then $w(M)\le 0.5$. Say $\lambda=e^{i\theta}\frac{1}{\sqrt{2}}$, if there is a point $z\in W(M)$ with $|z|>0.5$ applying a certain rotation to $B$ this point intersect the $y=0$ line contradicting the fact that the real part of $W(M)$ is in $ [-0.5;*[$.

Taking $B=\begin{pmatrix}0&\sqrt{2}\\0&0\end{pmatrix}$ and $S=\{z\in\mathbb{C}, |z|\le \frac{1}{\sqrt{2}}\}$ is a counterexample. The bound is $\sigma_1(B)\le \sqrt{2}$.

For any matrix $B,M\in \mathbb{M}_n(\mathbb{C})$ the numerical range of $B$ is the set $W(B)=\{x^*Bx, x\in \mathbb{C}^n, x^*x=1\}$. Also $W(\dfrac{M+M^*}{2})=\mathfrak{R}(W(M))$, $W(\lambda B)=\lambda W(B)$ (rotation and homothety) and $\sigma_1(M)\le 2w(M)$ where $w(M)$ is the numerical radius of $M$, (these are known), with $\lambda B=M$. We see that if $\dfrac{M+M^*}{2}\ge -0.5I$, then $w(M)\le 0.5$. Say $\lambda=e^{i\theta}\frac{1}{\sqrt{2}}$, if there is a point $z\in W(M)$ with $|z|>0.5$ applying a certain rotation to $B$ this point intersect the $y=0$ line contradicting the fact that the real part of $W(M)$ is in $ [-0.5;*[$.

Taking $B=\begin{pmatrix}0&\sqrt{2}\\0&0\end{pmatrix}$ and $S=\{z\in\mathbb{C}, |z|\le \frac{1}{\sqrt{2}}\}$ is a counterexample. The bound is $\sigma_1(B)\le \sqrt{2}$.

For any matrix $B,M\in \mathbb{M}_n(\mathbb{C})$ the numerical range of $B$ is the set $W(B)=\{x^*Bx, x\in \mathbb{C}^n, x^*x=1\}$. Also $W(\dfrac{M+M^*}{2})=\mathfrak{R}(W(M))$, $W(\lambda B)=\lambda W(B)$ (rotation and homothety) and $\sigma_1(M)\le 2w(M)$ where $w(M)$ is the numerical radius of $M$, (these are known), with $\lambda B=M$. We see that if $\dfrac{M+M^*}{2}\ge -0.5I$, for any $\lambda\in S$, then $w(M)\le 0.5$. Say $\lambda=e^{i\theta}\frac{1}{\sqrt{2}}$, if there is a point $z\in W(M)$ with $|z|>0.5$ applying a certain rotation to $B$ this point intersect the $y=0$ line contradicting the fact that the real part of $W(M)$ is in $ [-0.5;*[$.

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Toni Mhax
Toni Mhax
  • 785
  • 5
  • 13

Taking $B=\begin{pmatrix}0&\sqrt{2}\\0&0\end{pmatrix}$ and $S=\{z\in\mathbb{C}, |z|\le \frac{1}{\sqrt{2}}\}$ is a counterexample. The bound is $\sigma_1(B)\le \sqrt{2}$.

For any matrix $B,M\in \mathbb{M}_n(\mathbb{C})$ the numerical range of $B$ is the set $W(B)=\{x^*Bx, x\in \mathbb{C}^n, x^*x=1\}$. Also $W(\dfrac{M+M^*}{2})=\mathfrak{R}(W(M))$, $W(\lambda B)=\lambda W(B)$ (rotation and homothety) and $\sigma_1(M)\le 2w(M)$ where $w(M)$ is the spectralnumerical radius of $M$, (these are known), with $\lambda B=M$. We see that if $\dfrac{M+M^*}{2}\ge -0.5I$, then $w(M)\le 0.5$. Say $\lambda=e^{i\theta}\frac{1}{\sqrt{2}}$, if there is a point $z\in W(M)$ with $|z|>0.5$ applying a certain rotation to $B$ this point intersect the $y=0$ line contradicting the fact that the real part of $W(M)$ is in $ [-0.5;*[$.

Taking $B=\begin{pmatrix}0&\sqrt{2}\\0&0\end{pmatrix}$ and $S=\{z\in\mathbb{C}, |z|\le \frac{1}{\sqrt{2}}\}$ is a counterexample. The bound is $\sigma_1(B)\le \sqrt{2}$.

For any matrix $B,M\in \mathbb{M}_n(\mathbb{C})$ the numerical range of $B$ is the set $W(B)=\{x^*Bx, x\in \mathbb{C}^n, x^*x=1\}$. Also $W(\dfrac{M+M^*}{2})=\mathfrak{R}(W(M))$, $W(\lambda B)=\lambda W(B)$ (rotation and homothety) and $\sigma_1(M)\le 2w(M)$ where $w(M)$ is the spectral radius of $M$, (these are known), with $\lambda B=M$. We see that if $\dfrac{M+M^*}{2}\ge -0.5I$, then $w(M)\le 0.5$. Say $\lambda=e^{i\theta}\frac{1}{\sqrt{2}}$, if there is a point $z\in W(M)$ with $|z|>0.5$ applying a certain rotation to $B$ this point intersect the $y=0$ line contradicting the fact that the real part of $W(M)$ is in $ [-0.5;*[$.

Taking $B=\begin{pmatrix}0&\sqrt{2}\\0&0\end{pmatrix}$ and $S=\{z\in\mathbb{C}, |z|\le \frac{1}{\sqrt{2}}\}$ is a counterexample. The bound is $\sigma_1(B)\le \sqrt{2}$.

For any matrix $B,M\in \mathbb{M}_n(\mathbb{C})$ the numerical range of $B$ is the set $W(B)=\{x^*Bx, x\in \mathbb{C}^n, x^*x=1\}$. Also $W(\dfrac{M+M^*}{2})=\mathfrak{R}(W(M))$, $W(\lambda B)=\lambda W(B)$ (rotation and homothety) and $\sigma_1(M)\le 2w(M)$ where $w(M)$ is the numerical radius of $M$, (these are known), with $\lambda B=M$. We see that if $\dfrac{M+M^*}{2}\ge -0.5I$, then $w(M)\le 0.5$. Say $\lambda=e^{i\theta}\frac{1}{\sqrt{2}}$, if there is a point $z\in W(M)$ with $|z|>0.5$ applying a certain rotation to $B$ this point intersect the $y=0$ line contradicting the fact that the real part of $W(M)$ is in $ [-0.5;*[$.

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Toni Mhax
Toni Mhax
  • 785
  • 5
  • 13

Taking $B=\begin{pmatrix}0&\sqrt{2}\\0&0\end{pmatrix}$ and $S=\{z\in\mathbb{C}, |z|\le \frac{1}{\sqrt{2}}\}$ is a counterexample. The bound is $\sigma_1(B)\le \sqrt{2}$.

For any matrix $B,M\in \mathbb{M}_n(\mathbb{C})$ the numerical range of $B$ is the set $W(B)=\{x^*Bx, x\in \mathbb{C}^n, x^*x=1\}$. Also $W(\dfrac{M+M^*}{2})=\mathfrak{R}(W(M))$, $W(\lambda B)=\lambda W(B)$ (rotation and homothety) and $\sigma_1(M)\le 2w(M)$ where $w(M)$ is the spectral radius of $M$, (these are known), with $\lambda B=M$. We see that if $\dfrac{M+M^*}{2}\ge -0.5I$, then $w(M)\le 0.5$. Say $\lambda=e^{i\theta}\frac{1}{\sqrt{2}}$, if there is a point $z\in W(M)$ with $|z|>0.5$ applying a certain rotation to $B$ this point intersect the $y=0$ line contradicting the fact that the real part of $W(M)$ is in $ [-0.5;*[$.