Physlib.QuantumMechanics.DDimensions.Operators.SpectralTheory.Symmetric

For a linear partially defined operator $T$ on a complex Hilbert space $H$, the real numerical range $\Theta_{\mathrm{re}}(T)$ is the projection of the numerical range $\Theta(T)$ onto the real axis. It is defined as the set of real parts of the values in the numerical range: $$\Theta_{\mathrm{re}}(T) = \{ \mathrm{Re}(\langle x, Tx \rangle) \mid x \in \mathcal{D}(T), \|x\| = 1 \}$$ where $\mathcal{D}(T)$ is the domain of $T$, $\langle \cdot, \cdot \rangle$ is the complex inner product, and $\mathrm{Re}(z)$ denotes the real part of a complex number $z$.

For a symmetric partially defined linear operator $T$ on a complex Hilbert space $H$, the real numerical range $\Theta_{\mathrm{re}}(T)$ is equal to the image of the numerical range $\Theta(T)$ under the real part function $z \mapsto \mathrm{Re}(z)$. That is, $\Theta_{\mathrm{re}}(T) = \{ \mathrm{Re}(z) \mid z \in \Theta(T) \}$.

Let $T$ be a symmetric partially defined linear operator on a complex Hilbert space $H$. The real numerical range of the negative operator $-T$ is equal to the negative of the real numerical range of $T$, expressed as: $$\Theta_{\mathrm{re}}(-T) = -\Theta_{\mathrm{re}}(T)$$ where $\Theta_{\mathrm{re}}(T) = \{ \mathrm{Re}(\langle x, Tx \rangle) \mid x \in \mathcal{D}(T), \|x\| = 1 \}$ and $-\Theta_{\mathrm{re}}(T) = \{ -r \mid r \in \Theta_{\mathrm{re}}(T) \}$.

Let $T$ be a symmetric partially defined linear operator on a complex Hilbert space $H$. For any complex number $z$ in the numerical range $\Theta(T)$, the imaginary part of $z$ is zero, i.e., $\text{Im}(z) = 0$.

Let $T$ be a symmetric linear partially defined operator on a complex Hilbert space. The numerical range $\Theta(T)$, which is the set of values $\{\langle x, Tx \rangle \in \mathbb{C} \mid x \in \mathcal{D}(T), \|x\| = 1\}$, is a subset of the real axis $\mathbb{R} \subset \mathbb{C}$.

Let $T$ be a symmetric linear partially defined operator on a complex Hilbert space $H$. The numerical range $\Theta(T) \subseteq \mathbb{C}$ is equal to the image of the real numerical range $\Theta_{\mathrm{re}}(T) \subseteq \mathbb{R}$ under the standard inclusion map from $\mathbb{R}$ to $\mathbb{C}$. That is, $\Theta(T) = \{ r + 0i \in \mathbb{C} \mid r \in \Theta_{\mathrm{re}}(T) \}$.

For a symmetric linear partially defined operator $T$ on a complex Hilbert space, the closure of its numerical range $\Theta(T)$ is a subset of the real numbers $\mathbb{R} \subset \mathbb{C}$. That is, $\overline{\Theta(T)} \subseteq \mathbb{R}$.

Let $H$ be a complex Hilbert space and $T$ be a symmetric partially defined linear operator on $H$. For any complex number $z \in \mathbb{C}$ such that its imaginary part is non-zero ($\text{Im}(z) \neq 0$), $z$ belongs to the regularity domain of $T$. That is, there exists a constant $c > 0$ such that $\|Tx - zx\| \geq c \|x\|$ for all $x$ in the domain of $T$.

Let $T$ be a symmetric partially defined linear operator on a complex Hilbert space. The regularity domain of $T$ contains all complex numbers with a non-zero imaginary part. That is, the complement of the real line in the complex plane is a subset of the regularity domain: $$\mathbb{C} \setminus \mathbb{R} \subseteq \text{regularityDomain}(T)$$ where the regularity domain of $T$ is the set of complex numbers $z \in \mathbb{C}$ for which there exists a constant $c > 0$ such that $\|Tx - zx\| \geq c \|x\|$ for all $x$ in the domain of $T$.

Let $H$ be a complex Hilbert space and $T$ be a symmetric partially defined linear operator on $H$. Let $\Theta_{\mathrm{re}}(T)$ denote the real numerical range of $T$, defined as $\{ \mathrm{Re}(\langle x, Tx \rangle) \mid x \in D(T), \|x\| = 1 \}$. If $m \in \mathbb{R}$ is a lower bound for $\Theta_{\mathrm{re}}(T)$, then the interval $(-\infty, m)$ (viewed as a subset of the complex plane $\mathbb{C}$) is contained in the regularity domain of $T$. That is, for every $z \in \mathbb{C}$ such that $\mathrm{Im}(z) = 0$ and $\mathrm{Re}(z) < m$, there exists a constant $c > 0$ such that $\|Tx - zx\| \geq c \|x\|$ for all $x \in D(T)$.

Let $H$ be a complex Hilbert space and $T$ be a symmetric partially defined linear operator on $H$. Let $\Theta_{\mathrm{re}}(T)$ denote the real numerical range of $T$, defined as $\{ \langle x, Tx \rangle \mid x \in D(T), \|x\| = 1 \}$. If $m \in \mathbb{R}$ is an upper bound for $\Theta_{\mathrm{re}}(T)$, then the image of the open interval $(m, \infty)$ under the natural embedding into $\mathbb{C}$ is contained within the regularity domain of $T$. That is, for any real $z > m$, there exists $c > 0$ such that $\|Tx - zx\| \geq c \|x\|$ for all $x \in D(T)$.

Let $H$ be a complex Hilbert space and $T$ be a symmetric partially defined linear operator on $H$. The regularity domain of $T$ is connected if and only if it contains at least one real number, i.e., there exists $x \in \mathbb{R}$ such that $x$ (viewed as a complex number) is in the regularity domain of $T$.

Let $T$ be a symmetric partially defined linear operator on a complex Hilbert space $H$. Suppose that the real numerical range of $T$, defined as $\Theta_{\mathrm{re}}(T) = \{ \langle x, Tx \rangle \mid x \in \mathcal{D}(T), \|x\| = 1 \}$, is bounded below. Then the regularity domain of $T$, which is the set of complex numbers $z \in \mathbb{C}$ such that there exists a constant $c > 0$ satisfying $\|(T - zI)x\| \geq c \|x\|$ for all $x \in \mathcal{D}(T)$, is a connected set.

Let $H$ be a complex Hilbert space and $T$ be a symmetric partially defined linear operator on $H$ with domain $\mathcal{D}(T)$. If the real numerical range $\Theta_{\mathrm{re}}(T) = \{ \langle x, Tx \rangle \mid x \in \mathcal{D}(T), \|x\| = 1 \}$ is bounded from above, then the regularity domain of $T$ is connected. Here, the regularity domain is defined as the set of complex numbers $z \in \mathbb{C}$ such that there exists a constant $c > 0$ satisfying $\|(T - zI)x\| \geq c \|x\|$ for all $x \in \mathcal{D}(T)$.

Let $H$ be a complex Hilbert space and $T$ be a symmetric partially defined linear operator on $H$. The point spectrum $\sigma_p(T)$, which is the set of all eigenvalues of $T$, is a subset of the real numbers $\mathbb{R}$. That is, every eigenvalue $\lambda$ of a symmetric operator is real.