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\fancyhead[LO]{\textbf{Octonionic Heisenberg-Robertson-Schrodinger and  Maccone-Pati Uncertainty Principles}}
\fancyhead[RE]{\textbf{K. Mahesh Krishna}}


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{\bf\large{{Octonionic Heisenberg-Robertson-Schrodinger and Maccone-Pati Uncertainty Principles}}}\\
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\textbf{K. Mahesh Krishna}\\
School of Mathematics and Natural Sciences\\
Chanakya University Global Campus\\
NH-648, Haraluru Village\\
Devanahalli Taluk, 	Bengaluru  North District\\
Karnataka State, 562 110, India\\
Email: kmaheshak@gmail.com\\

Date: \today
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\textbf{Abstract}: 
Let $\mathcal{O}$ be an octonionic Hilbert space. 	Let  $A:	\mathcal{D}(A)\subseteq \mathcal{O}\to  \mathcal{O}$ and $B:\mathcal{D}(B)\subseteq \mathcal{O}\to  \mathcal{O}$  be possibly unbounded self-adjoint linear operators. 
Then for all $h \in  \mathcal{D}(AB)\cap  \mathcal{D}(BA)$ with $\|h\|=1$, we show that 
\begin{align*}
(1) \quad \quad \quad 	\Delta _h(A)\Delta _h(B)\geq 
\frac{\sqrt{|\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle|^2+|\langle [A,B]h, h \rangle |^2}}{2}
\end{align*}
and 
	\begin{align*}
(2) \quad \quad \quad 	\Delta _h(A)^2+	\Delta _h(B)^2 \geq \frac{|\langle (A+B)h, y \rangle|^2+|\langle (A-B)h, y \rangle|^2}{2}, \quad \forall y \in \mathcal{O} \text{ satisfying } \|y\|\leq1 \text{ and } \langle h, y \rangle =0, 
\end{align*}		
where $	\Delta _h(A)\coloneqq \|Ah-\langle Ah, h \rangle h \|$. We call Inequality (1) as octonionic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as octonionic Maccone-Pati uncertainty principle.  \\
\textbf{Keywords}:  Uncertainty principle, Octonionic Hilbert space.\\
\textbf{Mathematics Subject Classification (2020)}: 17A35, 46S10, 46C05.\\

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\section{Introduction}
Let $\mathcal{H}$ be a complex Hilbert space and $A$ be a possibly unbounded self-adjoint linear  operator defined on the domain $\mathcal{D}(A)\subseteq \mathcal{H}$. For $h \in \mathcal{D}(A)$ with $\|h\|=1$, define the \textbf{uncertainty} or \textbf{variance} of $A$ at the point $h$ as 
\begin{align*}
	\Delta _h(A)\coloneqq \|Ah-\langle Ah, h \rangle h \|=\sqrt{\|Ah\|^2-\langle Ah, h \rangle^2}. 
\end{align*}
In 1929, Robertson \cite{ROBERTSON} derived the following operator form of the uncertainty principle (also called  as uncertainty relation or uncertainty inequality) by  Heisenberg  in 1927 \cite{HEISENBERG}. Recall that for two linear operators $A:	\mathcal{D}(A)\subseteq \mathcal{H}\to  \mathcal{H}$ and $B:\mathcal{D}(B)\subseteq \mathcal{H}\to  \mathcal{H}$, we define the commutator $[A,B] \coloneqq AB-BA$ and the anti-commutator $\{A,B\}\coloneqq AB+BA$.
\begin{theorem} \cite{ROBERTSON, HEISENBERG, VONNEUMANNBOOK, DEBNATHMIKUSINSKI}  (\textbf{Heisenberg-Robertson Uncertainty Principle})
	Let  $A:\mathcal{D}(A)\subseteq \mathcal{H}\to  \mathcal{H}$ and $B:\mathcal{D}(B)\subseteq \mathcal{H}\to  \mathcal{H}$  be possibly unbounded self-adjoint linear operators. Then for all $h \in \mathcal{D}(AB)\cap  \mathcal{D}(BA)$ with $\|h\|=1$, we have 
	\begin{align}\label{HR}
		\frac{1}{2} \left(\Delta _h(A)^2+	\Delta _h(B)^2\right)\geq \frac{1}{4} \left(\Delta _h(A)+	\Delta _h(B)\right)^2 \geq  \Delta _h(A)	\Delta _h(B)   \geq  \frac{1}{2}|\langle [A,B]h, h \rangle |.
	\end{align}
\end{theorem}
In 1930,  Schrodinger improved Inequality (\ref{HR}) \cite{SCHRODINGER}. 
\begin{theorem} \cite{SCHRODINGER, BERTLMANNFRIIS} \label{SE}
	(\textbf{Heisenberg-Robertson-Schrodinger  Uncertainty Principle})
	Let  $A:	\mathcal{D}(A)\subseteq \mathcal{H}\to  \mathcal{H}$ and $B:	\mathcal{D}(B)\subseteq \mathcal{H}\to  \mathcal{H}$  be possibly unbounded self-adjoint linear operators. Then for all $h \in \mathcal{D}(AB)\cap  \mathcal{D}(BA)$ with $\|h\|=1$, we have 
	\begin{align*}
		\Delta _h(A)	\Delta _h(B)    &\geq |\langle Ah, Bh \rangle-\langle Ah, h \rangle \langle Bh, h \rangle|\\
		&=\frac{\sqrt{|\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle|^2+|\langle [A,B]h, h \rangle |^2}}{2}\\
		&=\frac{\sqrt{(\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle)^2-\langle [A,B]h, h \rangle ^2}}{2}.
	\end{align*}	
\end{theorem}
 In 2014,  Maccone and Pati derived the following  uncertainty principles which work for orthogonal vectors  \cite{MACCONEPATI}.
\begin{theorem}  \cite{MACCONEPATI} (\textbf{Maccone-Pati Uncertainty Principles}) \label{MP}
	Let  $A:	\mathcal{D}(A)\subseteq \mathcal{H}\to  \mathcal{H}$ and $B:	\mathcal{D}(B)\subseteq \mathcal{H}\to  \mathcal{H}$  be possibly unbounded self-adjoint linear operators. Then for all $h \in \mathcal{D}(AB)\cap  \mathcal{D}(BA)$ with $\|h\|=1$, we have 
	\begin{enumerate}[\upshape(i)]
		\item 
		\begin{align*}
			\Delta _h(A)^2+	\Delta _h(B)^2 &\geq|\langle (A+B)h, k \rangle|^2-\langle \{A,B\}h, h \rangle+2\langle Ah, h \rangle\langle Bh, h \rangle, \\
			&\quad \forall k \in \mathcal{H} \text{ satisfying } \|k\|=1 \text{ and } \langle h, k \rangle =0.
		\end{align*}
		\item 
		\begin{align*}
			\Delta _h(A)^2+	\Delta _h(B)^2 &\geq|\langle (A-B)h, k \rangle|^2+\langle \{A,B\}h, h \rangle-2\langle Ah, h \rangle\langle Bh, h \rangle, \\
			&\quad \forall k \in \mathcal{H} \text{ satisfying } \|k\|=1 \text{ and } \langle h, k \rangle =0.
		\end{align*}
		\item
		\begin{align*}
			\Delta _h(A)^2+	\Delta _h(B)^2 \geq \frac{|\langle (A+B)h, k \rangle|^2+|\langle (A-B)h, k \rangle|^2}{2}, \quad \forall k \in \mathcal{H} \text{ satisfying } \|k\|=1 \text{ and } \langle h, k \rangle =0.
		\end{align*}			
		\item 
		\begin{align*}
			\Delta _h(A)\Delta _h(B)  \geq	\frac{2\langle Ah, h \rangle\langle Bh, h \rangle-\langle \{A,B\}h, h \rangle}{2-\left|\left\langle \left(\frac{A}{\Delta _h(A)}+\frac{B}{\Delta _h(B)}\right)h, k\right\rangle \right|^2}, \quad \forall k \in \mathcal{H} \text{ satisfying } \|k\|=1 \text{ and } \langle h, k \rangle =0.
		\end{align*}
		\item 
		\begin{align*}
			\Delta _h(A)\Delta _h(B)  \geq	\frac{-2\langle Ah, h \rangle\langle Bh, h \rangle+\langle \{A,B\}h, h \rangle}{2-\left|\left\langle \left(\frac{A}{\Delta _h(A)}-\frac{B}{\Delta _h(B)}\right)h, k\right\rangle \right|^2}, \quad \forall k \in \mathcal{H} \text{ satisfying } \|k\|=1 \text{ and } \langle h, k \rangle =0.
		\end{align*}
	\end{enumerate}	
\end{theorem}










We derive octonionic  analogues of Theorem \ref{SE} in Theorem \ref{NH} and Theorem \ref{MP} in Theorem \ref{QMP}. We end the introduction by recalling the definition of octonionic Hilbert spaces.

Let  $\{e_j\}_{j=1}^7$ be the free symbols with multiplicative relations governed by the following table \cite{BAEZ}. 
\begin{table}[h!]
	\centering
	\renewcommand{\arraystretch}{1.3}
	\begin{tabular}{c|ccccccc}
		$\times$ & $e_1$ & $e_2$ & $e_3$ & $e_4$ & $e_5$ & $e_6$ & $e_7$ \\ \hline
		$e_1$ & $-1$ & $e_4$ & $e_7$ & $-e_2$ & $-e_6$ & $e_5$ & $-e_3$ \\
		$e_2$ & $-e_4$ & $-1$ & $e_5$ & $e_1$ & $-e_3$ & $e_7$ & $-e_6$ \\
		$e_3$ & $-e_7$ & $-e_5$ & $-1$ & $e_6$ & $e_2$ & $-e_4$ & $e_1$ \\
		$e_4$ & $e_2$ & $-e_1$ & $-e_6$ & $-1$ & $e_7$ & $e_3$ & $-e_5$ \\
		$e_5$ & $e_6$ & $e_3$ & $-e_2$ & $-e_7$ & $-1$ & $-e_1$ & $e_4$ \\
		$e_6$ & $-e_5$ & $-e_7$ & $e_4$ & $-e_3$ & $e_1$ & $-1$ & $e_2$ \\
		$e_7$ & $e_3$ & $e_6$ & $-e_1$ & $e_5$ & $-e_4$ & $-e_2$ & $-1$
	\end{tabular}
\end{table}\\

Let  
\begin{align*}
	\mathbb{O}\coloneqq \left\{a_0+\sum_{j=1}^{7}a_je_j: a_0, \dots, a_7 \in \mathbb{R} \right\}
\end{align*}
be the non-associative division algebra of octonion numbers with the conjugation
\begin{align*}
	\overline{a_0+\sum_{j=1}^{7}a_je_j}\coloneqq a_0-\sum_{j=1}^{7}a_je_j, \quad \forall a_0+\sum_{j=1}^{7}a_je_j \in 	\mathbb{O}
\end{align*}
and the modulus 
\begin{align*}
	\left|a_0+\sum_{j=1}^{7}a_je_j\right|\coloneqq \sqrt{\sum_{j=0}^{7}a_j^2}, \quad \forall a_0+\sum_{j=1}^{7}a_je_j \in 	\mathbb{O}.	
\end{align*}
\begin{definition}\cite{GOLDSTINEHORWITZ}
	A right octonionic  vector space 	 $\mathcal{O}$  over $\mathbb{O}$ is said to be a (right)  \textbf{octonionic Hilbert space} if there exists a  map (called as octonionic inner product) $ \langle \cdot, \cdot \rangle: \mathcal{O}\times \mathcal{O} \to \mathbb{O}$ such that the following hold. 
	\begin{enumerate}[\upshape(i)]
		\item $\langle x, x \rangle  \geq 0$, $\forall x \in \mathcal{O}$. If $x \in  \mathcal{O}$ satisfies $\langle x, x \rangle  =0 $, then $x=0$.
		\item $\langle x+y, z \rangle  =\langle x, z \rangle+\langle y, z \rangle$, $\forall x,y,z \in \mathcal{O}$.
		\item  $\operatorname{Re}(\langle x, yq \rangle)  =\operatorname{Re}(\langle x, y \rangle q)$, $\forall x,y \in \mathcal{O}$, $\forall q \in \mathbb{O}$.
		\item  $\langle x, yr \rangle  =\langle x, y \rangle r$, $\forall x,y \in \mathcal{O}$, $\forall r \in \mathbb{R}$.
		\item  $\langle x, xq \rangle  =\langle x, x \rangle q$, $\forall x \in \mathcal{O}$, $\forall q \in \mathbb{O}$.
		\item $\langle x, y \rangle=\overline{\langle y,x \rangle}$, $\forall x,y \in \mathcal{O}$.
		\item $\mathcal{O}$ is complete w.r.t. the norm $\|x\|\coloneqq \sqrt{\langle x, x \rangle}$,  $\forall x \in \mathcal{O}$.
	\end{enumerate}
\end{definition}


\section{Octonionic  Heisenberg-Robertson-Schrodinger Uncertainty Principle}
In this article,  $\mathcal{O}$ denotes a right octonionic Hilbert space. Throughout the entire paper, given $x \in \mathcal{O}$ and a  $r \in \mathbb{R}\setminus \{0\}$, the expression $x/r$ denotes $x r^{-1}$. Note that this is meaningful only for division by non-zero real numbers and not for arbitrary octonions, since the octonions are not commutative. 
Let $\mathcal{O}$ be  an octonionic Hilbert space and $A$ be a possibly unbounded self-adjoint linear operator  defined on domain $\mathcal{D}(A)\subseteq \mathcal{O}$. For $h \in  \mathcal{D}(A)$ with $\|h\| =1$, we define the \textbf{uncertainty} or \textbf{variance} of $A$ at the point $h$ as 
\begin{align*}
	\Delta _h(A)\coloneqq \|Ah-\langle Ah, h \rangle h \|=\sqrt{\|Ah\|^2-\langle Ah, h \rangle^2}. 
\end{align*}
Note that $\langle Ah, h \rangle \in \mathbb{R}$. For two maps $A:	\mathcal{D}(A)\subseteq \mathcal{O}\to  \mathcal{O}$ and $B:	\mathcal{D}(B)\subseteq \mathcal{O}\to  \mathcal{O}$, we define the commutator $[A,B] \coloneqq AB-BA$ and the anti-commutator  $\{A,B\}\coloneqq AB+BA$.
\begin{theorem}	(\textbf{Octonionic Heisenberg-Robertson-Schrodinger  Uncertainty Principles})\label{NH}
Let $\mathcal{O}$ be an octonionic Hilbert space. 	Let  $A:	\mathcal{D}(A)\subseteq \mathcal{O}\to  \mathcal{O}$ and $B:	\mathcal{D}(B)\subseteq \mathcal{O}\to  \mathcal{O}$  be possibly unbounded self-adjoint linear operators. Then for all $h \in  \mathcal{D}(AB)\cap  \mathcal{D}(BA)$ with $\|h\|=1$, we have 	
	\begin{enumerate}[\upshape(i)]
		\item 
		\begin{align*}
			\Delta _h(A)\Delta_h(B)\geq\frac{|\langle [A,B]h,h \rangle|}{2}.
		\end{align*}
	\item 
	\begin{align*}
		\Delta _h(A)\Delta_h(B)\geq    \frac{|\langle \{A,B\}h,h \rangle -2\langle Ah, h \rangle \langle Bh, h \rangle|}{2}.
	\end{align*}
\item 
\begin{align*}
\Delta _h(A)\Delta _h(B)&\geq \frac{\sqrt{|(\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle)^2-\langle [A,B]h, h \rangle ^2|}}{2}\\
&=\frac{\sqrt{|\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle|^2+|\langle [A,B]h, h \rangle |^2}}{2}.
\end{align*}
	\end{enumerate}
\end{theorem}
\begin{proof}
	Let $h \in  \mathcal{D}(AB)\cap  \mathcal{D}(BA)$ be such that  $\|h\|=1$. Since $A$ and $B$ are self-adjoint, we first make an important observation: $\langle Ah, h \rangle,\langle Bh, h \rangle \in \mathbb{R}$.
	\begin{enumerate}[\upshape(i)]
		\item 
		\begin{align*}
			\Delta _h(A)\Delta _h(B)&\geq\langle  Ah -\langle Ah, h \rangle h, Bh -\langle Bh, h \rangle h\rangle\\
			&= |\langle Ah, Bh \rangle-\langle Ah, h \rangle\langle Bh, h \rangle|\\
			&\geq \frac{|\langle Ah, Bh \rangle-\langle Ah, h \rangle\langle Bh, h \rangle-\overline{\langle Ah, Bh \rangle-\langle Ah, h \rangle\langle Bh, h \rangle}|}{2}\\
			&=\frac{|\langle Ah, Bh \rangle-\langle Ah, h \rangle\langle Bh, h \rangle-\langle Bh, Ah \rangle+\langle Bh, h \rangle\langle Ah, h \rangle|}{2}\\
			&=\frac{|\langle BAh, h \rangle-\langle ABh, h \rangle|}{2}\\
			&=\frac{|-\langle (AB-BA)h,h \rangle |}{2}\\
			&=\frac{|\langle [A,B]h,h \rangle|}{2}.
		\end{align*}
	\item 
	\begin{align*}
		\Delta _h(A)\Delta _h(B)&\geq\langle  Ah -\langle Ah, h \rangle h, Bh -\langle Bh, h \rangle h\rangle\\
		&= |\langle Ah, Bh \rangle-\langle Ah, h \rangle\langle Bh, h \rangle|\\
		&\geq \frac{|\langle Ah, Bh \rangle-\langle Ah, h \rangle\langle Bh, h \rangle+\overline{\langle Ah, Bh \rangle-\langle Ah, h \rangle\langle Bh, h \rangle}|}{2}\\
		&=\frac{|\langle Ah, Bh \rangle-\langle Ah, h \rangle\langle Bh, h \rangle+\langle Bh, Ah \rangle-\langle Bh, h \rangle\langle Ah, h \rangle|}{2}\\
		&=\frac{|\langle BAh, h \rangle-\langle Ah, h \rangle\langle Bh, h \rangle+\langle ABh, h \rangle-\langle Bh, h \rangle\langle Ah, h \rangle|}{2}\\
		&=\frac{|\langle (AB+BA)h,h \rangle - 2\langle Ah, h \rangle\langle Bh, h \rangle|}{2}\\
		&=\frac{|\langle \{A,B\}h,h \rangle -2\langle Ah, h \rangle \langle Bh, h \rangle|}{2}.
	\end{align*}
	\item  Define 
	\begin{align*}
		z\coloneqq Ah -\langle Ah, h \rangle h, \quad w\coloneqq Bh -\langle Bh, h \rangle h.
	\end{align*}
	Then 
	\begin{align*}
		\langle z, w\rangle+ \langle w, z\rangle&=\langle  Ah -\langle Ah, h \rangle h, Bh -\langle Bh, h \rangle h\rangle+\langle  Bh -\langle Bh, h \rangle h, Ah -\langle Ah, h \rangle h\rangle\\
		&=\langle BAh, h \rangle-\langle Ah, h \rangle\langle Bh, h \rangle+\langle ABh, h \rangle-\langle Bh, h \rangle\langle Ah, h \rangle\\
		&=\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle
	\end{align*}
	and 
	\begin{align*}
		\langle z, w\rangle- \langle w, z\rangle&=\langle  Ah -\langle Ah, h \rangle h, Bh -\langle Bh, h \rangle h\rangle-\langle  Bh -\langle Bh, h \rangle h, Ah -\langle Ah, h \rangle h\rangle\\
		&=\langle BAh, h \rangle-\langle Ah, h \rangle\langle Bh, h \rangle-\langle ABh, h \rangle+\langle Bh, h \rangle\langle Ah, h \rangle\\
		&=-\langle [A,B]h, h \rangle.
	\end{align*}
Therefore 
\begin{align*}
	&\Delta _h(A)^2\Delta_h(B)^2+\Delta _h(B)^2\Delta_h(A)^2\geq |\langle z, w\rangle|^2+|\langle w, z\rangle|^2\\
	&=\frac{(\langle z, w\rangle+ \langle w, z\rangle)^2-(\langle z, w\rangle- \langle w, z\rangle)^2}{2}\\
	&=\frac{(\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle)^2-\langle [A,B]h, h \rangle^2}{2}.
\end{align*}
 Note that 
 \begin{align*}
 	\overline{\langle [A,B]h, h \rangle }=-\langle [A,B]h, h \rangle 
 \end{align*}
and 
\begin{align*}
	\overline{\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle}=\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle.
\end{align*}
 Therefore 
 \begin{align*}
 	-\langle [A,B]h, h \rangle ^2=|\langle [A,B]h, h \rangle |^2
 \end{align*}
 and 
\begin{align*}
	(\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle)^2=|\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle|^2.
\end{align*}
	\end{enumerate}
\end{proof}

\section{Octonionic  Maccone-Pati Uncertainty Principles}
Here we derive octonionic Maccone-Pati uncertainty principles.
\begin{theorem}   (\textbf{Octonionic Maccone-Pati Uncertainty Principles}) \label{QMP}
Let $\mathcal{O}$ be an octonionic Hilbert space.	Let  $A:	\mathcal{D}(A)\subseteq \mathcal{O}\to  \mathcal{O}$ and $B:	\mathcal{D}(B)\subseteq \mathcal{O}\to  \mathcal{O}$  be possibly unbounded self-adjoint linear operators. Then for all $h \in  \mathcal{D}(AB)\cap  \mathcal{D}(BA)$ with $\|h\|=1$, we have 
	\begin{enumerate}[\upshape(i)]
		\item 
		\begin{align*}
			\Delta _h(A)^2+	\Delta _h(B)^2 &\geq|\langle (A+B)h, y \rangle|^2-\langle \{A,B\}h, h \rangle+2\langle Ah, h \rangle\langle Bh, h \rangle, \\
			&\quad \forall y \in \mathcal{O} \text{ satisfying } \|y\|\leq 1 \text{ and } \langle h,y \rangle =0.
		\end{align*}
		\item 
		\begin{align*}
			\Delta _h(A)^2+	\Delta _h(B)^2 &\geq|\langle (A-B)h, y \rangle|^2+\langle \{A,B\}h, h \rangle-2\langle Ah, h \rangle\langle Bh, h \rangle, \\
			&\quad \forall y \in \mathcal{O} \text{ satisfying } \|y\|\leq 1 \text{ and } \langle h,y \rangle =0.
		\end{align*}
		\item
		\begin{align*}
			\Delta _h(A)^2+	\Delta _h(B)^2 \geq \frac{|\langle (A+B)h, y \rangle|^2+|\langle (A-B)h, y \rangle|^2}{2}, \quad \forall y \in \mathcal{O} \text{ satisfying } \|y\|\leq1 \text{ and } \langle h, y \rangle =0.
		\end{align*}	
		\item 
		\begin{align*}
			\Delta _h(A)\Delta _h(B)  \geq	\frac{2\langle Ah, h \rangle\langle Bh, h \rangle-\langle \{A,B\}h, h \rangle}{2-\left|\left\langle \left(\frac{A}{\Delta _h(A)}+\frac{B}{\Delta _h(B)}\right)h, y\right\rangle \right|^2}, \quad \forall y \in \mathcal{O} \text{ satisfying } \|y\|\leq1 \text{ and } \langle h,y \rangle =0.
		\end{align*}
		\item 
		\begin{align*}
			\Delta _h(A)\Delta _h(B)  \geq	\frac{-2\langle Ah, h \rangle\langle Bh, h \rangle+\langle \{A,B\}h,h \rangle}{2-\left|\left\langle \left(\frac{A}{\Delta _h(A)}-\frac{B}{\Delta _h(B)}\right)h, y\right\rangle \right|^2}, \quad \forall y \in \mathcal{O} \text{ satisfying }  \|y\|\leq1\text{ and } \langle h, y \rangle =0.
		\end{align*}
	\end{enumerate}	
\end{theorem}
\begin{proof}
		Let $h \in  \mathcal{D}(AB)\cap  \mathcal{D}(BA)$ with $\|h\|=1$. Let  	$y \in \mathcal{O}$ be such that $ \|y\|\leq 1 $ and $ \langle h,y \rangle =0.$
		\begin{enumerate}[\upshape(i)]
			\item 
				\begin{align*}
				&|\langle (A+B)h, y \rangle|^2=|\langle Ah-\langle Ah, h \rangle h+Bh-\langle Bh, h \rangle h, y \rangle|^2\\
				&\leq \|y\|^2\langle  (Ah-\langle Ah, h \rangle h)+ (Bh-\langle Bh, h \rangle h), (Ah-\langle Ah, h \rangle h)+ (Bh-\langle Bh, h \rangle h)\rangle\\
				&\leq \langle Ah-\langle Ah, h \rangle h, Ah-\langle Ah, h \rangle h\rangle +\langle Bh-\langle Bh, h \rangle h, Bh-\langle Bh, h \rangle h\rangle \\
				&\quad +\langle Bh-\langle Bh, h \rangle h, Ah-\langle Ah, h \rangle h\rangle+\langle Ah-\langle Ah, h \rangle h, Bh-\langle Bh, h \rangle h\rangle\\
				&=\Delta_h(A)^2+\Delta_h(B)^2+\langle Bh, Ah \rangle -\langle Bh, h \rangle\langle Ah, h \rangle+\langle Ah, Bh \rangle -\langle Ah, h \rangle\langle Bh, h \rangle\\
				&=\Delta_h(A)^2+\Delta_h(B)^2+\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle \langle Bh, h \rangle.
			\end{align*}
			\item Similar to (i).
			\item We get by adding (i) and (ii).
			\item 
				\begin{align*}
				&\left|\left\langle \left(\frac{A}{\Delta_h(A)}+\frac{B}{\Delta_h(B)}\right)h, y\right \rangle \right|^2=\left|\left\langle \frac{Ah-\langle Ah, h \rangle h}{\Delta_h(A)}+\frac{Bh-\langle Bh, h \rangle h}{\Delta_h(B)}, y\right \rangle \right|^2\\
				&\leq \|y\|^2\left\langle \frac{Ah-\langle Ah, h \rangle h}{\Delta_h(A)}+\frac{Bh-\langle Bh, h \rangle h}{\Delta_h(B)}, \frac{Ah-\langle Ah, h \rangle h}{\Delta_h(A)}+\frac{Bh-\langle Bh, h \rangle h}{\Delta_h(B)}\right \rangle\\
				&\leq \left\langle \frac{Ah-\langle Ah, h \rangle h}{\Delta_h(A)},  \frac{Ah-\langle Ah, h \rangle h}{\Delta_h(A)}\right \rangle+\left\langle \frac{Bh-\langle Bh, h \rangle h}{\Delta_h(B)},  \frac{Bh-\langle Bh, h \rangle h}{\Delta_h(B)}\right \rangle\\
				&\quad +\left\langle \frac{Bh-\langle Bh, h \rangle h}{\Delta_h(B)},  \frac{Ah-\langle Ah, h \rangle h}{\Delta_h(A)}\right \rangle+\left\langle \frac{Ah-\langle Ah, h \rangle h}{\Delta_h(A)},  \frac{Bh-\langle Bh, h \rangle h}{\Delta_h(B)}\right \rangle\\
				&=1+1+\frac{\langle Bh, Ah \rangle-\langle Bh, h \rangle\langle Ah, h \rangle}{\Delta_h(A)\Delta_h(B)}+\frac{\langle Ah, Bh \rangle-\langle Ah, h \rangle\langle Bh, h \rangle}{\Delta_h(A)\Delta_h(B)}\\
				&=2+\frac{-2\langle Ah, h \rangle\langle Bh, h \rangle +\langle \{A,B\}h, h \rangle}{\Delta_h(A)\Delta_h(B)}.
			\end{align*}
			Therefore 
			\begin{align*}
				&-\frac{-2\langle Ah, h \rangle\langle Bh, h \rangle +\langle \{A,B\}h, h \rangle}{\Delta_h(A)\Delta_h(B)}\leq 2-\left|\left\langle \left(\frac{A}{\Delta_h(A)}+\frac{B}{\Delta_h(B)}\right)h, y\right \rangle \right|^2\\	
				&\implies \frac{2\langle Ah, h \rangle\langle Bh, h \rangle -\langle \{A,B\}h, h \rangle}{2-\left|\left\langle \left(\frac{A}{\Delta_h(A)}+\frac{B}{\Delta_h(B)}\right)h, y\right \rangle \right|^2}\leq \Delta_h(A)\Delta_h(B).
			\end{align*}
		\item Similar to (iv).
		\end{enumerate}
\end{proof}
\section{Acknowledgments}
AI is used for the literature survey and grammar improvement.






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\end{document}