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\fancyhead[LO]{\textbf{NONCOMMUTATIVE BUZANO-DRAGOMIR INEQUALITY AND APPLICATIONS}}
\fancyhead[RE]{\textbf{K. MAHESH KRISHNA}}


\begin{document}
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{\bf{Noncommutative Buzano-Dragomir Inequality and Applications}}\\
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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}: Dragomir [\textit{Bull. Aust. Math. Soc., 2016}] showed that the Buzano inequality holds for orthogonal projections on Hilbert spaces.  Dragomir [\textit{Linear Multilinear Algebra, 2016}] also derived the most general form of the Buzano inequality for bounded linear operators on Hilbert spaces. We show that Dragomir result extends to adjointable morphisms on Hilbert C*-modules. Using this generalization, we derive bounds for the roots of polynomials over commutative unital C*-algebras. We formulate the notion of noncommutative numerical range and derive numerical radius bounds for self-adjoint morphisms on Hilbert C*-modules over  unital C*-algebras. We formulate several open problems, including the  noncommutative Toeplitz-Hausdorff, von Neumann inequality, Ando inequality, Berger power dilation, Kittaneh inequality, Crouzeix problems. 


\textbf{Keywords}: Buzano Inequality, Numerical radius, Polynomial roots, Hilbert C*-modules, Crouzeix inequality.

\textbf{Mathematics Subject Classification (2020)}: 47A12, 46L08.\\

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\section{Introduction}
More than half-century ago, Buzano derived the following generalization of the \textit{Cauchy-Schwarz inequality} \cite{BUZANO}.
\begin{theorem}  \cite{FUJII, BUZANO, STEELE} \label{BT}   (\textbf{Buzano Inequality})
	Let $\mathcal{H}$ be a complex Hilbert space (with the inner product linear in the first variable and conjugate linear in the second variable). Then 
	\begin{align}\label{BI}
		|\langle \tau, h\rangle \langle h, \omega\rangle |\leq \frac{\|h\|^2(|\langle \tau, \omega \rangle|+\|\tau\|\|\omega\|)}{2}\leq \|h\|^2\|\tau\|\|\omega\|, \quad \forall h, \tau, \omega \in \mathcal{H}.
	\end{align}
In other words, 
\begin{align*}
	\left|\left\langle \tau, \frac{h}{\|h\|}\right\rangle \left \langle \frac{h}{\|h\|}, \omega \right\rangle\right|\leq \frac{|\langle \tau, \omega \rangle|+\|\tau\|\|\omega\|}{2}\leq \|\tau\|\|\omega\|, \quad \forall  \tau, \omega \in \mathcal{H}, \forall h \in \mathcal{H}\setminus \{0\}.
\end{align*}
Further, we have following.
\begin{enumerate}[\upshape(i)]
	\item Let $\tau, \omega \in \mathcal{H}\setminus\{0\}$ be such that $\langle \tau, \omega \rangle =0$. Choose $\alpha, \beta \in \mathbb{C}$ with $|\beta|=1$. Define 
	\begin{align*}
		h_{\alpha, \beta}\coloneqq \alpha \left(\frac{\tau}{\|\tau\|}+\beta\frac{\omega}{\|\omega\|}\right).
	\end{align*}
Then 
\begin{align*}
		|\langle \tau, 	h_{\alpha, \beta}\rangle \langle 	h_{\alpha, \beta}, \omega\rangle |= \frac{\|	h_{\alpha, \beta}\|^2(|\langle \tau, \omega \rangle|+\|\tau\|\|\omega\|)}{2}.
\end{align*}
	\item  Let $\tau, \omega \in \mathcal{H}\setminus\{0\}$ be such that $\langle \tau, \omega \rangle \neq 0$. For $\alpha \in \mathbb{C}$, define 
	\begin{align*}
		h_{\alpha}\coloneqq \alpha \left(\frac{\tau}{\|\tau\|}+\frac{\langle \tau, \omega \rangle}{|\langle \tau, \omega \rangle |}\frac{\omega}{\|\omega\|}\right).
	\end{align*}
	Then 
	\begin{align*}
		|\langle \tau, 	h_{\alpha}\rangle \langle 	h_{\alpha}, \omega\rangle |= \frac{\|	h_{\alpha}\|^2(|\langle \tau, \omega \rangle|+\|\tau\|\|\omega\|)}{2}.
	\end{align*}
\end{enumerate}
\end{theorem} 
Ten years ago,  Dragomir made the following far reaching generalization of Theorem \ref{BT} \cite{DRAGOMIR2}.
\begin{theorem}   \cite{DRAGOMIR2} \label{DT} (\textbf{Buzano-Dragomir Inequality})
	Let $\mathcal{H}$ be a complex Hilbert space and  $P:\mathcal{H}\to \mathcal{H}$ be an orthogonal projection. Then 
	\begin{align}\label{DI}
		|\langle P\tau, \omega\rangle |\leq \frac{|\langle \tau, \omega \rangle|+\|\tau\|\|\omega\|}{2}, \quad \forall \tau, \omega \in \mathcal{H}.	
	\end{align}
\end{theorem}
Now note that for any nonzero $h\in \mathcal{H}$, the map 
\begin{align*}
	\mathcal{H} \ni \tau \mapsto \left\langle \tau, \frac{h}{\|h\|} \right \rangle \frac{h}{\|h\|} \in 	\mathcal{H}
\end{align*}
is a rank one orthogonal projection.  Thus Inequality (\ref{DI}) recovers Inequality (\ref{BI}). 
Inequality (\ref{BI}) has  various applications, we describe few.  Recall that \cite{GUSTAFSONRAO, HALMOS, HORNJOHNSON, GAUWUBOOK, STONEBOOK} the \textbf{numerical range} of a bounded linear operator $T:\mathcal{H} \to \mathcal{H} $ is defined as 
\begin{align*}
	W_\mathbb{C}(T)\coloneqq \{\langle Th, h\rangle: h \in \mathcal{H}, \|h\|=1\}
\end{align*}
and the \textbf{numerical radius} of  $T$ is defined as 
\begin{align*}
	w_\mathbb{C}(T)\coloneqq \sup_{h \in \mathcal{H}, \|h\|=1}|\langle Th, h\rangle|.
\end{align*}
For  $ \tau, \omega \in \mathcal{H}$, define the rank one operator  
\begin{align*}
	\tau \otimes \omega : \mathcal{H} \ni h \mapsto (\tau \otimes \omega)h \coloneqq \langle h, \omega \rangle \tau \in \mathcal{H}.
\end{align*}
In 1993, Fujii and Kubo derived following result, using Inequality (\ref{BI}) \cite{FUJII}.
\begin{theorem} \cite{FUJII} \label{FKR}
	Let $\mathcal{H}$ be a complex Hilbert space and $ \tau, \omega \in \mathcal{H}$. Then 
	\begin{align*}
		w_\mathbb{C}(\tau \otimes \omega)=\frac{|\langle \tau, \omega \rangle|+\|\tau\|\|\omega\|}{2}.
	\end{align*}
\end{theorem}
Let $p(z)\coloneqq a_0+a_1z+\cdots +a_{n-1}z^{n-1}+z^n \in \mathbb{C}[z]$. A direct observation reveals that the zeros of $p$ are the eigenvalues of the \textbf{Frobenius companion matrix} 
\begin{align*}
	C_p\coloneqq 
	\begin{pmatrix}
		-a_{n-1}&-a_{n-2}&-a_{n-3}&\cdots &-a_2 &-a_1&-a_0\\
		1&0&0&\cdots &0 &0&0\\
		0&1&0&\cdots &0 &0&0\\
		\vdots &\vdots &\vdots & & \vdots &\vdots &\vdots\\
		0&0&0&\cdots &0 &0&0\\
		0&0&0&\cdots &1 &0&0\\
	 	0&0&0&\cdots &0 &1&0\\
	\end{pmatrix} \in \mathbb{M}_n(\mathbb{C})
\end{align*}
and the eigenvalues of $C_p$ are the same as the zeros of $p$. Let $R$ be the right shift matrix defined by 
\begin{align*}
	R\coloneqq 
	\begin{pmatrix}
	     0&0&0&\cdots &0 &0&0\\
		1&0&0&\cdots &0 &0&0\\
		0&1&0&\cdots &0 &0&0\\
		\vdots &\vdots &\vdots & & \vdots &\vdots &\vdots\\
		0&0&0&\cdots &0 &0&0\\
		0&0&0&\cdots &1 &0&0\\
		0&0&0&\cdots &0 &1&0\\
	\end{pmatrix} \in \mathbb{M}_n(\mathbb{C}).
\end{align*}
Define 
\begin{align*}
	{\bf{a}}\coloneqq \begin{pmatrix}
		\overline{a_{n-1}}\\
		\overline{a_{n-2}}\\
			\overline{a_{n-3}}\\
		\vdots \\
			\overline{a_{2}}\\
		\overline{a_{1}}\\
		\overline{a_{0}}\\
	\end{pmatrix}, \quad 
	{\bf{e_1}} \coloneqq \begin{pmatrix}
1\\
0\\
0\\
	\vdots \\
	0\\
	0\\
0\\
\end{pmatrix}. 
\end{align*}
Then 
\begin{align}\label{FK}
	C_p=R-	{\bf{e_1}}\otimes {\bf{a}}.
\end{align}
Fujii and Kubo derived following bound for the zeros of $p$ using Inequality \ref{BI} and the representation (\ref{FK}) \cite{FUJII}.
\begin{theorem} \cite{FUJII} \label{FKT} (\textbf{Fujii-Kubo polynomial root upper bound})
	Let $p(z)\coloneqq a_0+a_1z+\cdots +a_{n-1}z^{n-1}+z^n \in \mathbb{C}[z]$.  If $\lambda$ is a  zero of $p$, then 
	\begin{align}\label{FKRB}
		|\lambda|&\leq w_\mathbb{C}(R)+\frac{\sqrt{\sum_{j=0}^{n-1}|a_j|^2}+|a_{n-1}|}{2}=\cos\left(\frac{\pi}{n+1}\right)+\frac{\sqrt{\sum_{j=0}^{n-1}|a_j|^2}+|a_{n-1}|}{2}.
	\end{align}
\end{theorem}
Let $p(z)\coloneqq a_0+a_1z+\cdots +a_{n-1}z^{n-1}+z^n \in \mathbb{C}[z]$ with $a_0\neq 0$. Define 
\begin{align*}
	q(z)\coloneqq \frac{1}{a_0}z^np\left(\frac{1}{z}\right)=\frac{1}{a_0}+\frac{a_{n-1}}{a_0}z+\cdots +\frac{a_{1}}{a_0}z^{n-1}+z^n \in  \mathbb{C}[z].
\end{align*}
We note that $\lambda \in \mathbb{C}\setminus \{0\}$ satisfies $p(\lambda)=0$ if and only if $q(1/\lambda)=0$ \cite{HORNJOHNSON}. Frobenius companion matrix of $q$ is 
\begin{align*}
	C_q\coloneqq 
	\begin{pmatrix}
		-\frac{a_1}{a_0}&-\frac{a_2}{a_0}&-\frac{a_3}{a_0}&\cdots &-\frac{a_{n-1}}{a_0} &-\frac{a_{n-1}}{a_0}&-\frac{1}{a_0}\\
		1&0&0&\cdots &0 &0&0\\
		0&1&0&\cdots &0 &0&0\\
		\vdots &\vdots &\vdots & & \vdots &\vdots &\vdots\\
		0&0&0&\cdots &0 &0&0\\
		0&0&0&\cdots &1 &0&0\\
		0&0&0&\cdots &0 &1&0\\
	\end{pmatrix} \in \mathbb{M}_n(\mathbb{C})
\end{align*}
By applying Theorem \ref{FKT} to $q$, we get following result. 
\begin{theorem} \cite{FUJII} (\textbf{Fujii-Kubo polynomial root lower bound})
	Let $p(z)\coloneqq a_0+a_1z+\cdots +a_{n-1}z^{n-1}+z^n \in \mathbb{C}[z]$ with $a_0 \neq 0$.  If $\lambda$ is a  zero of $p$, then 	
	\begin{align*}
		|\lambda| \geq \frac{2|a_0|}{2|a_0|\cos\left(\frac{\pi}{n+1}\right)+\sqrt{1+\sum_{j=1}^{n-1}|a_j|^2}+|a_{1}|}.
	\end{align*}
\end{theorem}
For any bounded linear operator $T:\mathcal{H} \to \mathcal{H} $, the Cauchy-Schwarz inequality gives
\begin{align}\label{NR}
		w_\mathbb{C}(T)\leq \|T\|.
\end{align} 
In 2007,  Dragomir strengthened Inequality (\ref{NR})  using Buzano inequality \cite{DRAGOMIR}.
\begin{theorem} \cite{DRAGOMIR} \label{DIT} (\textbf{Dragomir Numerical Radius Inequality}) Let $T:\mathcal{H} \to \mathcal{H} $ be a bounded linear operator. Then
	\begin{align*}
		w_\mathbb{C}(T) \leq \frac{1}{\sqrt{2}}\sqrt{w_\mathbb{C}(T^2)+\|T\|^2}\leq \|T\|.
	\end{align*}
\end{theorem}

Our fundamental motivation comes from the following question: What is the noncommutative analogue of Theorem \ref{DT}? This is then naturally connected with the notion of Hilbert C*-modules which  are first introduced by Kaplansky \cite{KAPLANSKY} for modules over commutative C*-algebras and later developed for modules over arbitrary C*-algebras by Paschke  \cite{PASCHKE} and Rieffel \cite{RIEFFEL}. 
\begin{definition} \cite{KAPLANSKY, PASCHKE, RIEFFEL}
	Let $\mathcal{A}$ be a  unital C*-algebra. A right module 	$\mathcal{E}$  over $\mathcal{A}$ is said to be a \textbf{semi-inner product C*-module} if there exists a  map $ \langle \cdot, \cdot \rangle: \mathcal{E}\times \mathcal{E} \to \mathcal{A}$ such that the following hold. 
	\begin{enumerate}[\upshape(i)]
		\item $\langle x, x \rangle  \geq 0$, $\forall x \in \mathcal{E}$. 
		\item $\langle x+y, z \rangle  =\langle x, z \rangle+\langle y, z \rangle$, $\forall x,y,z \in \mathcal{E}$.
		\item  $\langle x, ya \rangle  =\langle x, y \rangle a$, $\forall x,y \in \mathcal{E}$, $\forall a \in \mathcal{A}$.
		\item $\langle x, y \rangle=\langle y,x \rangle^*$, $\forall x,y \in \mathcal{E}$.
	\end{enumerate}
\end{definition}
Semi-inner product C*-modules satisfy noncommutative Cauchy-Schwarz inequality. 
\begin{theorem}  \cite{LANCE} \label{NCS} (\textbf{Kaplansky-Paschke-Rieffel Inequality})
	Let $\mathcal{E}$ be a semi-inner product C*-module. Then 
	\begin{align*}
		\langle y, x \rangle \langle x, y \rangle \leq \|\langle x, x \rangle\| \langle y, y \rangle, \quad \forall x, y \in \mathcal{E}.
	\end{align*}
In particular, 
	\begin{align*}
	\|\langle y, x \rangle\|^2 \leq \|\langle x, x \rangle\| \|\langle y, y \rangle\|, \quad \forall x, y \in \mathcal{E}.
\end{align*}
\end{theorem}
\begin{definition} \cite{LANCE}
A semi-inner product C*-module $\mathcal{E}$	is said to be an \textbf{inner product C*-module} if $ x \in \mathcal{E}$ satisfies $\langle x, x \rangle  = 0$, then $x=0$.  
\end{definition}	
\begin{definition} \cite{LANCE}
A inner product C*-module $\mathcal{E}$	is said to be \textbf{Hilbert C*-module} if 	$\mathcal{E}$ is complete w.r.t. the norm $\|x\|\coloneqq \sqrt{\|\langle x, x \rangle\|}$,  $\forall x \in \mathcal{E}$.
\end{definition}	
In 2012, Khosravi, Drnovsek and Moslehian derived noncommutative version of Inequality (\ref{BI}) \cite{KHOSRAVI}. 
\begin{theorem} \cite{KHOSRAVI} \label{KDMT} (\textbf{Buzano-Khosravi-Drnovsek-Moslehian Inequality})
	Let $\mathcal{E}$ be a Hilbert C*-module. Then 
	\begin{align}\label{KI}
			\|\langle \omega, x\rangle \langle x, \tau\rangle \|\leq \frac{\|\langle \tau, \omega \rangle\|+\|\tau\|\|\omega\|}{2}\leq \|\tau\|\|\omega\|, \quad \forall \tau, \omega \in \mathcal{E}, \forall x \in \mathcal{E} \text{ with } \langle x, x \rangle =1.
	\end{align}
\end{theorem}
In this article, we show that Theorem \ref{DT} extends to orthogonal projections on Hilbert C*-modules (which will also generalize Theorem \ref{KDMT}).  We also derive modular analogue of Theorems \ref{FKT} and \ref{DIT}.




\section{Noncommutative  Buzano-Dragomir  Inequality}
We start by deriving noncommutative analogue of Theorem \ref{DT}. 
\begin{theorem}\label{PC}(\textbf{Noncommutative Buzano-Dragomir Inequality})
		Let $\mathcal{E}$ be a  Hilbert C*-module over a unital C*-algebra $\mathcal{A}$ and  $P:\mathcal{E}\to \mathcal{E}$ be an orthogonal projection.  Then 
	\begin{align*}
		\|\langle \tau, P\omega\rangle \|\leq \frac{\|\langle \tau, \omega \rangle\|+\|\tau\|\|\omega\|}{2}, \quad \forall \tau, \omega \in \mathcal{E}.	
	\end{align*}
\end{theorem}
\begin{proof}
We find 
\begin{align*}
	\|2\langle \tau, P\omega\rangle-\langle \tau, \omega \rangle\|&=\| \langle \tau, 2P\omega-\omega\rangle\|\leq \|\tau\|\|2P\omega-\omega\|.
\end{align*}	
We now find 
\begin{align*}
\|2P\omega-\omega\|^2&=\|\langle 2P\omega-\omega, 2P\omega-\omega\rangle \|\\
&=\|4\langle P \omega, P\omega\rangle-2\langle P \omega, \omega\rangle-2\langle  \omega, P\omega\rangle+\langle \omega, \omega\rangle\|\\
&=\|\langle \omega, \omega\rangle\|\\
&=\|\omega\|^2.
\end{align*}
Therefore 
\begin{align*}
	2\|\langle \tau, P\omega\rangle\|-\|\langle \tau, \omega \rangle\|	&\leq \|2\langle \tau, P\omega\rangle-\langle \tau, \omega \rangle\|\leq \|\tau\|\|\omega\|.
\end{align*}
\end{proof}
\begin{corollary}
Theorem \ref{KDMT} follows from Theorem \ref{PC}.
\end{corollary}
\begin{proof}
	Let $\mathcal{E}$ be a  Hilbert C*-module and $ x \in \mathcal{E}$ with $\langle x, x \rangle =1$. Define 
	\begin{align*}
		P: \mathcal{E} \ni y \mapsto Py\coloneqq x \langle x, y \rangle \in \mathcal{E}.
	\end{align*}
Then $P$ is an orthogonal projection. Theorem \ref{PC} gives 
\begin{align*}
	\|\langle \tau, x\rangle \langle x, \omega\rangle \|&=\|\langle \tau,  x \langle x, \omega  \rangle \rangle \|=\|\langle \tau, P\omega \rangle\| \\
	&\leq \frac{\|\langle \tau, \omega \rangle\|+\|\tau\|\|\omega\|}{2}, \quad \forall \tau, \omega \in \mathcal{E}.	
\end{align*}
\end{proof}
In 2016, Dragomir further improved Theorem \ref{DT} for bounded linear operators \cite{DRAGOMIR3}.
\begin{theorem} \cite{DRAGOMIR3} \label{DIG} (\textbf{Buzano-Dragomir Inequality})
	Let $\mathcal{H}$ be a complex Hilbert space and  $T:\mathcal{H}\to \mathcal{H}$ be a bounded linear operator. Then 
	\begin{align*}
		|\langle T\tau, T\omega\rangle |\leq \frac{\|T\|^2(|\langle \tau, \omega \rangle|+\|\tau\|\|\omega\|)}{2}, \quad \forall \tau, \omega \in \mathcal{H}.	
	\end{align*}	
\end{theorem}
We now derive following noncommutative version of Theorem \ref{DIG}. 
\begin{theorem}  \label{NDI}   (\textbf{Noncommutative Buzano-Dragomir Inequality})
	Let $\mathcal{E}$ be a  Hilbert C*-module over a unital C*-algebra $\mathcal{A}$ and  $T:\mathcal{E}\to \mathcal{E}$ be an adjointable morphism.  Then
	\begin{align*}
		\|\langle T\tau, T\omega\rangle \|\leq \|T\|^2\|\langle \tau, \omega \rangle\|+\bigg\|\langle \tau, \tau \rangle\|T\|^2-\langle T\tau, T\tau \rangle\bigg\|^\frac{1}{2}\bigg\|\langle \omega, \omega \rangle\|T\|^2-\langle T\omega, T\omega \rangle\bigg\|^\frac{1}{2},  \quad \forall \tau, \omega \in \mathcal{E}.
	\end{align*}
\end{theorem}
\begin{proof}
Our proof is motivated by the Hilbert space argument due to Dragomir \cite{DRAGOMIR3}.	Let $T:\mathcal{E}\to \mathcal{E}$ be an adjointable morphism and $\tau, \omega \in \mathcal{E}$. Define
	\begin{align*}
		[\cdot, \cdot]:  \mathcal{E} \times  \mathcal{E} \ni (x,y)\mapsto [x,y]\coloneqq \langle x, y \rangle \|T\|^2-\langle Tx, Ty \rangle \in \mathcal{A}.
	\end{align*}
	Then $ [\cdot, \cdot]$ is a semi-inner  product. By applying Theorem \ref{NCS} to this semi-inner product, we get
	\begin{align*}
			\big\|[x,y]	\big\|^2\leq \|[x, x]\|\|[y,y]\|, \quad \forall x, y \in \mathcal{E}.
	\end{align*}
	In particular,
	\begin{align*}
		\left\|[\tau, \omega]\right\|^2\leq \|[\tau, \tau]\|\|[\omega, \omega]\|.
	\end{align*}
	Previous inequality gives
	\begin{align*}
		\bigg\|\langle \tau, \omega \rangle \|T\|^2-\langle T\tau, T\omega \rangle\bigg\|^2 \leq \bigg\|\langle \tau, \tau \rangle\|T\|^2-\langle T\tau, T\tau \rangle\bigg\|\bigg\|\langle \omega, \omega \rangle\|T\|^2-\langle T\omega, T\omega \rangle\bigg\|.
	\end{align*}
Therefore 
\begin{align*}
		\|\langle T\tau, T\omega\rangle\| - \|T\|^2\|\langle \tau, \omega \rangle\|&\leq 	\bigg\|\langle \tau, \omega \rangle \|T\|^2-\langle T\tau, T\omega \rangle\bigg\|\\
		&\leq \bigg\|\langle \tau, \tau \rangle\|T\|^2-\langle T\tau, T\tau \rangle\bigg\|^\frac{1}{2}\bigg\|\langle \omega, \omega \rangle\|T\|^2-\langle T\omega, T\omega \rangle\bigg\|^\frac{1}{2}.
\end{align*}
\end{proof}
 We are unable to simplify Theorem \ref{NDI}. We therefore ask following problem. 
 \begin{problem}
 	Let $\mathcal{E}$ be a  Hilbert C*-module over a unital C*-algebra $\mathcal{A}$ and  $T:\mathcal{E}\to \mathcal{E}$ be an adjointable morphism.  Whether
 	\begin{align*}
 		\|\langle T\tau, T\omega\rangle \|\leq \frac{\|T\|^2(\|\langle \tau, \omega \rangle\|+\|\tau\|\|\omega\|)}{2}, \quad \forall \tau, \omega \in \mathcal{E}?
 	\end{align*}
 \end{problem}
Let $\mathcal{E}$ be a Hilbert C*-module over a unital C*-algebra $\mathcal{A}$. 
Similar to Hilbert space case, we define the \textbf{noncommutative numerical range} of an adjointable morphism $T:\mathcal{E}\to \mathcal{E}$ is defined as 
\begin{align*}
	W_\mathcal{A}(T)\coloneqq \{\langle x, Tx\rangle: x \in \mathcal{E}, \langle x, x \rangle =1\}\subseteq \mathcal{A}
\end{align*}
and the \textbf{noncommutative numerical radius} $T$ is defined as 
\begin{align*}
w_\mathcal{A}(T)\coloneqq \sup_{x \in \mathcal{E}, \langle x, x \rangle =1}\|\langle x, Tx\rangle\|\geq 0.
\end{align*}
Note that our definition of numerical range and radius on Hilbert C*-modules differ from the existing same notion on Hilbert C*-modules \cite{ZAMANI, MEHRAZINAMYARIOMIDVAR, RAJIC}
We give various examples. In all following examples, $\mathcal{A}$ is a   unital C*-algebra.
\begin{example}
Let $b \in \mathcal{A}$. Then 
	\begin{align*}
		W_\mathcal{A}(b)=\{a^*ba: a\in \mathcal{A}, a^*a=1\}
	\end{align*}
	and 
	\begin{align*}
		w_\mathcal{A}(b)&=\sup\{\|a^*ba\|: a\in \mathcal{A}, a^*a=1\}\\
		&= \|b\|.
	\end{align*}
If $\mathcal{A}$ is commutative, then $	W_\mathcal{A}(b)=\{b\}$. 
\end{example}
\begin{example}
	Let $d\in \mathbb{N}$ and $I_d$ be the identity matrix of size $d$ by $d$. Then we have 
	\begin{align*}
		W_\mathcal{A}(I_d)=\{\langle x, x\rangle: x \in \mathcal{A}^d, \langle x, x \rangle=1\}=\left\{1\right\}
	\end{align*}
	and 
	\begin{align*}
		w_\mathcal{A}(I_d)=1.
	\end{align*}
\end{example}
\begin{example}
	Define 
	\begin{align*}
		M\coloneqq \begin{pmatrix}
			0 & 1  \\
			0 & 0 \\
		\end{pmatrix}.
	\end{align*}
	Then 
	\begin{align*}
		W_\mathcal{A}(M)&=\{\langle x, Mx\rangle: x \in \mathcal{A}^2, \langle x, x \rangle=1\}\\
		&=\left\{a^*b:a, b \in \mathcal{A}, a^*a+b^*b=1\right\}
	\end{align*}
	and 
	\begin{align*}
		w_\mathcal{A}(M)&=\sup\left\{\|a^*b\|:a, b \in \mathcal{A}, a^*a+b^*b=1\right\}.
	\end{align*}
We note that 
\begin{align*}
	a^*bb^*a\leq \frac{a^*a+b^*b}{2}, \quad \forall a, b \in \mathcal{A}. 
\end{align*}
Therefore 
	\begin{align*}
	w_\mathcal{A}(M)&=\sup\left\{\|a^*b\|:a, b \in \mathcal{A}, a^*a+b^*b=1\right\}\\
	&\leq \frac{1}{\sqrt{2}}\sup\{\sqrt{\|a^*a+b^*b\|}:a, b \in \mathcal{A}, a^*a+b^*b=1\}\\
	&= \frac{1}{\sqrt{2}}.
\end{align*}
\end{example}
\begin{example}
	Let $d\in \mathbb{N}$ and $b_1, \dots, b_d \in \mathcal{A}$. Define $\operatorname{diag}((b_j)_{j=1}^d)$ as the diagonal matrix with diagonal entries $b_1, \dots, b_d$. Then 
	\begin{align*}
		W_\mathcal{A}(\operatorname{diag}((b_j)_{j=1}^d))=\left\{\sum_{j=1}^da_j^*b_ja_j: (a_j)_{j=1}^d\in \mathcal{A}^d, \sum_{j=1}^da_j^*a_j=1\right\}
	\end{align*}
	and 
	\begin{align*}
		w_\mathcal{A}(\operatorname{diag}((b_j)_{j=1}^d))&=\sup\left\{\left\|\sum_{j=1}^da_j^*b_ja_j\right\|: (a_j)_{j=1}^d\in 	\mathcal{A}^d, \sum_{j=1}^da_j^*a_j=1\right\}\\
		&\leq \sum_{j=1}^d\|b_j\|.
	\end{align*}
The set $	W_\mathcal{A}(\operatorname{diag}((b_j)_{j=1}^d))$ is the  C*-convex hull of $\{b_j\}_{j=1}^d$ \cite{LOEBLPAULSEN}.
\end{example}
\begin{example}
	Let $R \in \mathbb{M}_{d}(\mathcal{A})$ be the right shift matrix defined by 
	\begin{align*}
		R\coloneqq \begin{pmatrix}
			0 & 0 & 0& \cdots & 0 & 0 & 0\\
			1 & 0 & 0& \cdots & 0 & 0 & 0\\
				0 & 1 & 0& \cdots & 0 & 0 & 0\\
			\vdots &\vdots  &  \vdots & & 	\vdots &\vdots & \vdots\\
				0 & 0 & 0& \cdots & 0 & 0 & 0\\
			0 & 0 & 0& \cdots & 1 & 0 & 0\\
			0& 0 & 0& \cdots & 0 & 1 & 0\\
		\end{pmatrix}.
	\end{align*}
	Thus the action of $R$ is 
	\begin{align*}
		R(a_j)_{j=1}^d=(0, a_1, a_2, \dots, a_{d-1}), \quad \forall (a_j)_{j=1}^d \in \mathcal{A}^d.
	\end{align*}
	We then have 
	\begin{align*}
		\langle  (a_j)_{j=1}^d, R(a_j)_{j=1}^d\rangle =a_2^*a_1+\cdots+ a_d^*a_{d-1}, \quad \forall (a_j)_{j=1}^d \in \mathcal{A}^d.
	\end{align*}
	Hence 
	\begin{align*}
		\|\langle (a_j)_{j=1}^d, R(a_j)_{j=1}^d\rangle\|&=\|a_2^*a_1+\cdots+ a_d^*a_{d-1}\|\\
		&\leq \left\|\sum_{j=2}^da_j^*a_j\right\|^\frac{1}{2}\left\|\sum_{k=1}^{d-1}a_k^*a_k\right\|^\frac{1}{2}\\
		&\leq \left\|\sum_{j=1}^da_j^*a_j\right\|^\frac{1}{2}\left\|\sum_{k=1}^{d}a_k^*a_k\right\|^\frac{1}{2}\\
		&=\left\|\sum_{j=1}^da_j^*a_j\right\|,  \quad \forall (a_j)_{j=1}^d \in \mathcal{A}^d.
	\end{align*}
		Therefore 
	\begin{align*}
		W_\mathcal{A}(R)=\left\{a_2^*a_1+\cdots+ a_d^*a_{d-1}:(a_j)_{j=1}^d \in \mathcal{A}^d, \sum_{j=1}^{d}a_j^*a_j=1\right\}
	\end{align*}
	and 
	\begin{align*}
		w_\mathcal{A}(R)&=\sup\left\{\|a_2^*a_1+\cdots+ a_d^*a_{d-1}\|:(a_j)_{j=1}^d \in \mathcal{A}^d, \sum_{j=1}^{d}a_j^*a_j=1\right\}\\
		&\leq 1.
	\end{align*}
\end{example}
\begin{example}
	Let $\mathcal{A}$ be a unital C*-algebra and $\ell^2(\mathcal{A})$ be the standard Hilbert C*-module defined by 
	\begin{align*}
	\ell^2(\mathcal{A})\coloneqq \left\{\{a_n\}_{n=1}^\infty: a_n \in \mathcal{A}, \forall n \in \mathbb{N}, \sum_{n=1}^{\infty}a_n^*a_n\in \mathcal{A}\right\}
	\end{align*}
equipped with inner product 
\begin{align*}
	\langle \{a_n\}_{n=1}^\infty, \{b_n\}_{n=1}^\infty\rangle \coloneqq \sum_{n=1}^{\infty}a_n^*b_n
, \quad \forall \{a_n\}_{n=1}^\infty, \{b_n\}_{n=1}^\infty \in \ell^2(\mathcal{A})
\end{align*}
and norm 
\begin{align*}
	\|\{a_n\}_{n=1}^\infty\|=\left\|\sum_{n=1}^{\infty}a_n^*a_n\right\|^\frac{1}{2}, \quad \forall \{a_n\}_{n=1}^\infty \in \ell^2(\mathcal{A}).
\end{align*}
Let $L$ be the left shift morphism defined by 
\begin{align*}
	L: \ell^2(\mathcal{A}) \ni \{a_n\}_{n=1}^\infty \mapsto L \{a_n\}_{n=1}^\infty\coloneqq \{a_{n+1}\}_{n=1}^\infty\in \ell^2(\mathcal{A}).
\end{align*}
Then 
	\begin{align*}
	W_\mathcal{A}(L)=\left\{\sum_{n=1}^{\infty}a_n^*a_{n+1}:\{a_n\}_{n=1}^\infty \in \ell^2(\mathcal{A}), \sum_{n=1}^{\infty}a_n^*a_{n}=1\right\}
\end{align*}
and 
\begin{align*}
	w_\mathcal{A}(L)&=\sup\left\{\left\|\sum_{n=1}^{\infty}a_n^*a_{n+1}\right\|:\{a_n\}_{n=1}^\infty \in \ell^2(\mathcal{A}), \sum_{n=1}^{\infty}a_n^*a_{n}=1\right\}\\
	&\leq 1.
\end{align*}
\end{example}
\begin{example}
	Define 
	\begin{align*}
		A\coloneqq \begin{pmatrix}
			a & b  \\
			c & d \\
		\end{pmatrix}.
	\end{align*}
	Then 
	\begin{align*}
		W_\mathcal{A}(A)&=\{x^*ax+x^*by+y^*cx+y^*dy: x, y \in \mathcal{A}, x^*x+y^*y=1\}
	\end{align*}
	and 
	\begin{align*}
		w_\mathcal{A}(A)&=\sup\{\|x^*ax+x^*by+y^*cx+y^*dy\|: x, y \in \mathcal{A}, x^*x+y^*y=1\}\\
		&\leq\|a\|+\|b\|+\|c\|+\|d\|.
	\end{align*} 
\end{example}





Given $r>0$ and $v \in \mathcal{A}$, the closed disc centered at $v \in \mathcal{A}$ of radius $ r$ is defined as 
\begin{align*}
	\mathbb{D}_{ \mathcal{A}}[v, r]\coloneqq \{z \in \mathcal{A}: \|z-v\|\leq r\}.
\end{align*}
Following are basic properties of noncommutative numerical range and radius. 
\begin{proposition}
Let $\mathcal{E}$ be a Hilbert C*-module over a unital C*-algebra $\mathcal{A}$. Let $\mathcal{A}^+$ be the set of all positive elements in $\mathcal{A}$. Let 	$T, S :\mathcal{E}\to \mathcal{E}$ be adjointable morphisms and $a,b \in \mathcal{A}$. Let $I$ be the identity morphism on $\mathcal{E}$.
	\begin{enumerate}[\upshape(i)]
	\item $W_\mathcal{A}(T)\subseteq \mathbb{D}_{\mathcal{A}}[0, \|T\|]$. 
	\item $0\leq  w_\mathcal{A}(T)\leq \|T\|$. 
	\item $W_\mathcal{A}(T+S)\subseteq W_\mathcal{A}(T)+W_\mathcal{A}(S)$. 
	\item $w_\mathcal{A}(S+T)\leq w_\mathcal{A}(S)+w_\mathcal{A}(T)$. 
	\item $W_\mathcal{A}(Ta)=W_\mathcal{A}(T)a$. 
	\item $w_\mathcal{A}(Ta)\leq \|a\|w_\mathcal{A}(T)$. 
	\item $W_\mathcal{A}(T^*)=(W_\mathcal{A}(T))^*$.
	\item $w_\mathcal{A}(T^*)=w_\mathcal{A}(T)$. 
	\item $W_\mathcal{A}(Ta+bI)=W_\mathcal{A}(T)a+b$.
		\item $W_\mathcal{A}(\operatorname{Re}(T))=\operatorname{Re}(W_\mathcal{A}(T))$ and  $W_\mathcal{A}(\operatorname{Im}(T))=\operatorname{Im}(W_\mathcal{A}(T))$.
		\item If $T\geq0$, then $W_\mathcal{A}(T)\subseteq \mathcal{A}^+$.
		\item Let $\mathcal{E}_0$ be a Hilbert C*-module over a unital C*-algebra $\mathcal{A}$ such that $\mathcal{E}$ is orthogonally complementable in $\mathcal{E}_0$. If an adjointable morphism $V:\mathcal{E}_0\to \mathcal{E}_0$  is a dilation of $T$, then $W_\mathcal{A}(T)\subseteq W_\mathcal{A}(V)$. 
			\item Let  $\mathcal{E}_1$ be an  orthogonally complementable closed submodule in $\mathcal{E}$. Let $P:\mathcal{E}\to \mathcal{E}_1$  be an onto orthogonal projection.  Then $W_\mathcal{A}(PT_{|\mathcal{E}_1})\subseteq W_\mathcal{A}(T)$. 
	\item $W_\mathcal{A}(U^*TU)=W_\mathcal{A}(T)$  for every unitary morphism $U:\mathcal{E}\to \mathcal{E}$. 
	\item Let $a \in \mathcal{A}$ be such that there exists a $x \in \mathcal{E}$ with $Tx=xa$ and $\langle x, x \rangle=1$. Then $a \in W_\mathcal{A}(T)$. In other words, 
	\begin{align*}
\sigma_1(T)\coloneqq	\{a \in \mathcal{A}: \exists x \in  \mathcal{E}, Tx=xa, \langle x, x \rangle=1\} \subseteq	W_\mathcal{A}(T)
	\end{align*}
and 
\begin{align*}
	\|a\|\leq w_\mathcal{A}(T), \quad \forall a \in \sigma_1(T).
\end{align*}
	\item 
	\begin{align*}
		w_\mathcal{A}(ST+TS)\leq\frac{w_\mathcal{A}((S+T)^2)+w_\mathcal{A}((S-T)^2)}{2}. 
	\end{align*}
In particular, if $ST=TS$, then
	\begin{align*}
	w_\mathcal{A}(ST)\leq\frac{w_\mathcal{A}((S+T)^2)+w_\mathcal{A}((S-T)^2)}{4}. 
\end{align*} 
	\item If a sequence $\{T_n\}_{n=1}^\infty $ of adjointable morphisms on $\mathcal{E}$ converges to an adjointable morphism  $T $ on $\mathcal{E}$ in the morphism norm, then the  sequence $\{w_\mathcal{A}(T_n)\}_{n=1}^\infty$ converges to $w_\mathcal{A}(T)$ in the norm.
\end{enumerate}	
\end{proposition}
Using generalized polarization identity, it is known  that $w_\mathbb{C}(T)\geq \|T\|/2$  \cite{GUSTAFSONRAO}. We are unable to derive noncommutative version of this result. 
For  $ \tau, \omega \in \mathcal{E}$, define  the morphism 
\begin{align*}
	\tau \otimes \omega : \mathcal{E} \ni x \mapsto (\tau \otimes \omega)x \coloneqq \tau \langle \omega, x \rangle \in \mathcal{E}.
\end{align*}
Following is modular version of Theorem \ref{FKR}.
\begin{theorem} 
	Let $\mathcal{E}$ be a   Hilbert C*-module over a commutative unital C*-algebra $\mathcal{A}$ and $ \tau, \omega \in \mathcal{E}$. Then 
	\begin{align}\label{RE}
		w_\mathcal{A}(\tau \otimes \omega)\leq \frac{\|\langle \tau, \omega \rangle\|+\|\tau\|\|\omega\|}{2}\leq \|\tau\|\|\omega\|.
	\end{align}
In particular, 
\begin{align*}
		w_\mathcal{A}(\tau \otimes \tau)\leq \|\tau\|^2.
\end{align*}
\end{theorem}
\begin{proof}
Let $ \tau, \omega \in \mathcal{E}$. Let $x \in \mathcal{E}$	with $\langle x, x \rangle=1$. Then using commutativity of C*-algebra,  
\begin{align*}
	\|\langle x, (\tau \otimes \omega)x \rangle\| &=\|\langle x, \tau \langle \omega, x \rangle\rangle \|=\|\langle x, \tau \rangle \langle \omega, x \rangle \|\\
	&=\|\langle \omega, x  \rangle \langle  x, \omega\rangle\|\leq \frac{\|\langle \tau, \omega \rangle\|+\|\tau\|\|\omega\|}{2}.
\end{align*}
Therefore 
\begin{align*}
		w_\mathcal{A}(\tau \otimes \omega)\leq \frac{\|\langle \tau, \omega \rangle\|+\|\tau\|\|\omega\|}{2}.
\end{align*} 
\end{proof}
Unlike the Hilbert space case, we are unable to derive equality in Inequality (\ref{RE}).
Let $\mathcal{A}$ be a unital C*-algebra and $p(z)\coloneqq a_0+a_1z+\cdots +a_{n-1}z^{n-1}+z^n \in \mathcal{A}[z]$. Define the \textbf{modular Frobenius companion matrix} 
\begin{align*}
	C_p\coloneqq 
	\begin{pmatrix}
		-a_{n-1}&-a_{n-2}&-a_{n-3}&\cdots &-a_2 &-a_1&-a_0\\
		1&0&0&\cdots &0 &0&0\\
		0&1&0&\cdots &0 &0&0\\
		\vdots &\vdots &\vdots & & \vdots &\vdots &\vdots\\
		0&0&0&\cdots &0 &0&0\\
		0&0&0&\cdots &1 &0&0\\
		0&0&0&\cdots &0 &1&0\\
	\end{pmatrix} 
\in \mathbb{M}_n(\mathcal{A}).
\end{align*}
Given $ n \in \mathbb{N}$, we consider $\mathcal{A}^n$ with the standard inner product 
\begin{align*}
	\langle (a_j)_{j=1}^n, (b_j)_{j=1}^n\rangle \coloneqq \sum_{j=1}^{n}a_j^*b_j, \quad \forall (a_j)_{j=1}^n, (b_j)_{j=1}^n \in \mathcal{A}^n. 
\end{align*} 
Hence the norm on $\mathcal{A}^n$ is 
\begin{align*}
	\|(a_j)_{j=1}^n\|=\left\|\sum_{j=1}^{n}a_j^*a_j\right\|^\frac{1}{2}, \quad \forall (a_j)_{j=1}^n \in \mathcal{A}^n.
\end{align*}
We then have the following result. 
\begin{theorem}
Let $\mathcal{A}$ be a commutative unital C*-algebra and $p(z)\coloneqq a_0+a_1z+\cdots +a_{n-1}z^{n-1}+z^n \in \mathcal{A}[z]$. If $b \in \mathcal{A}$ satisfies $p(b)=0$, then $b \in W_\mathcal{A}(C_p)$. 
\end{theorem}
\begin{proof}
Define 
\begin{align*}
	x \coloneqq 
	\begin{pmatrix}
		b^{n-1}\\
		b^{n-2}\\
			b^{n-3}\\
		\vdots\\
		b^2\\
		b\\
		1
	\end{pmatrix} 
\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2}.
\end{align*}
Then $\langle x, x \rangle =1$. Using commutativity of C*-algebra, we have 

\begin{align*}
&\langle x, C_p x\rangle=\\
&\left\langle 	\begin{pmatrix}
	b^{n-1}\\
	b^{n-2}\\
		b^{n-3}\\
	\vdots\\
	b^2\\
	b\\
	1
\end{pmatrix}\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2}, 	
	\begin{pmatrix}
		-a_{n-1}&-a_{n-2}&-a_{n-3}&\cdots &-a_2 &-a_1&-a_0\\
		1&0&0&\cdots &0 &0&0\\
		0&1&0&\cdots &0 &0&0\\
		\vdots &\vdots &\vdots & & \vdots &\vdots &\vdots\\
		0&0&0&\cdots &0 &0&0\\
		0&0&0&\cdots &1 &0&0\\
		0&0&0&\cdots &0 &1&0\\
	\end{pmatrix}
	\begin{pmatrix}
	b^{n-1}\\
	b^{n-2}\\
		b^{n-3}\\
	\vdots\\
		b^2\\
	b\\
	1
\end{pmatrix} \left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2}
\right\rangle\\
&=\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2} \left\langle \begin{pmatrix}
	b^{n-1}\\
	b^{n-2}\\
		b^{n-3}\\
	\vdots\\
		b^2\\
	b\\
	1
\end{pmatrix},
\begin{pmatrix}
	-a_{n-1}&-a_{n-2}&-a_{n-3}&\cdots &-a_2 &-a_1&-a_0\\
	1&0&0&\cdots &0 &0&0\\
	0&1&0&\cdots &0 &0&0\\
	\vdots &\vdots &\vdots & & \vdots &\vdots &\vdots\\
	0&0&0&\cdots &0 &0&0\\
	0&0&0&\cdots &1 &0&0\\
	0&0&0&\cdots &0 &1&0\\
\end{pmatrix}
\begin{pmatrix}
	b^{n-1}\\
	b^{n-2}\\
		b^{n-3}\\
	\vdots\\
		b^2\\
	b\\
	1
\end{pmatrix} 
\right\rangle\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2} \\
&=\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2} \left\langle \begin{pmatrix}
	b^{n-1}\\
	b^{n-2}\\
		b^{n-3}\\
	\vdots\\
		b^2\\
	b\\
	1
\end{pmatrix},
\begin{pmatrix}
	-\sum_{j=0}^{n-1}a_{j}b^j\\\
	b^{n-1}\\
		b^{n-2}\\
	\vdots\\
		b^3\\
	b^2\\
	b
\end{pmatrix} 
\right\rangle\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2}\\
&=\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2} \left\langle \begin{pmatrix}
	b^{n-1}\\
	b^{n-2}\\
		b^{n-3}\\
	\vdots\\
		b^2\\
	b\\
	1
\end{pmatrix},
\begin{pmatrix}
	b^n\\\
	b^{n-1}\\
		b^{n-2}\\
	\vdots\\
		b^3\\
	b^2\\
	b
\end{pmatrix} 
\right\rangle\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2}\\
  &=\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2} \left\langle \begin{pmatrix}
  	b^{n-1}\\
  	b^{n-2}\\
  		b^{n-3}\\
  	\vdots\\
  		b^2\\
  	b\\
  	1
  \end{pmatrix},
  \begin{pmatrix}
  	b^{n-1}\\\
  	b^{n-2}\\
  		b^{n-3}\\
  	\vdots\\
  		b^2\\
  	b\\
  	1
  \end{pmatrix} 
  \right\rangle b \left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2}\\
  &=\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2}\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)b \left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2}\\
  &=\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2}\left(\sum_{j=0}^{n-1}(b^*)^jb^j\right) \left(\sum_{j=0}^{n-1}(b^*)^jb^j\right)^\frac{-1}{2}b\\
  &=b.
\end{align*}
Therefore $b=\langle x, C_p x\rangle \in W_\mathcal{A}(C_p)$. 
\end{proof}
 We now derive  modular analogue of Theorem \ref{FKT}.
 \begin{theorem}
 	Let $\mathcal{A}$ be a commutative unital C*-algebra and $p(z)\coloneqq a_0+a_1z+\cdots +a_{n-1}z^{n-1}+z^n \in \mathcal{A}[z]$. If $b \in \mathcal{A}$ satisfies $p(b)=0$, then 
 		\begin{align*}
 		\|b\|&\leq w_\mathcal{A}(R)+\frac{\sqrt{\left\|\sum_{j=0}^{n-1}a_j^*a_j\right\|}+\|a_{n-1}\|}{2}\\
 		 &\leq w_\mathcal{A}(R)+\frac{\sqrt{\sum_{j=0}^{n-1}\|a_j\|^2}+\|a_{n-1}\|}{2}, 
 	\end{align*}
 where   $R$ is the right shift matrix defined by 
 \begin{align*}
 	R\coloneqq 
 	\begin{pmatrix}
 		0&0&0&\cdots &0 &0&0\\
 		1&0&0&\cdots &0 &0&0\\
 		0&1&0&\cdots &0 &0&0\\
 		\vdots &\vdots &\vdots & & \vdots &\vdots &\vdots\\
 		0&0&0&\cdots &0 &0&0\\
 		0&0&0&\cdots &1 &0&0\\
 		0&0&0&\cdots &0 &1&0\\
 	\end{pmatrix} \in \mathbb{M}_n(\mathcal{A}).
 \end{align*}
 \end{theorem}
 \begin{proof}
Define 
\begin{align*}
	{\bf{a}}\coloneqq \begin{pmatrix}
		(a_{n-1})^*\\
	(a_{n-2})^*\\
	(a_{n-3})^*\\
		\vdots \\
	(a_2)^*\\
		(a_1)^*\\
			(a_0)^*\\
	\end{pmatrix} \in \mathcal{A}^n, \quad 
	{\bf{e_1}} \coloneqq \begin{pmatrix}
		1\\
		0\\
		0\\
		\vdots \\
		0\\
		0\\
		0\\
	\end{pmatrix} \in \mathcal{A}^n.
\end{align*}
Then 
\begin{align}\label{QFK}
	C_p=R-{\bf{e_1}}\otimes {\bf{a}}.
\end{align}
Using (\ref{QFK}) and Inequality (\ref{RE}), we get 
	\begin{align*}
	\|b\|&\leq w_\mathcal{A}(C_p)\\
	&=w_\mathcal{A}(R-{\bf{e_1}}\otimes 	{\bf{a}}) \\
	&\leq w_\mathcal{A}(R)+w_\mathcal{A}({\bf{e_1}}\otimes {\bf{a}})\\
	&\leq w_\mathcal{A}(R)+\frac{\sqrt{\left\|\sum_{j=0}^{n-1}a_j^*a_j\right\|}+\|a_{n-1}\|}{2}\\
	&\leq w_\mathcal{A}(R)+\frac{\sqrt{\sum_{j=0}^{n-1}\|a_j\|^2}+\|a_{n-1}\|}{2}.
	\end{align*}
\end{proof}
	Let $\mathcal{A}$ be a commutative unital C*-algebra. Let $p(z)\coloneqq a_0+a_1z+\cdots +a_{n-1}z^{n-1}+z^n \in \mathcal{A}[z]$ with $a_0$ invertible. Define 
\begin{align*}
	q(z)\coloneqq  \frac{1}{a_0}z^np\left(\frac{1}{z}\right)=\frac{1}{a_0}+\frac{a_{n-1}}{a_0}z+\cdots +\frac{a_{1}}{a_0}z^{n-1}+z^n \in  \mathcal{A}[z].
\end{align*}
We note that an invertible element $b \in \mathcal{A}$ satisfies $p(b)=0$ if and only if $q(1/b)=0$. Frobenius companion matrix of $q$ is 
\begin{align*}
	C_q\coloneqq 
	\begin{pmatrix}
		-\frac{a_1}{a_0}&-\frac{a_2}{a_0}&-\frac{a_3}{a_0}&\cdots &-\frac{a_{n-1}}{a_0} &-\frac{a_{n-1}}{a_0}&-\frac{1}{a_0}\\
		1&0&0&\cdots &0 &0&0\\
		0&1&0&\cdots &0 &0&0\\
		\vdots &\vdots &\vdots & & \vdots &\vdots &\vdots\\
		0&0&0&\cdots &0 &0&0\\
		0&0&0&\cdots &1 &0&0\\
		0&0&0&\cdots &0 &1&0\\
	\end{pmatrix} \in \mathbb{M}_n(\mathcal{A}).
\end{align*}
By applying Theorem \ref{FKRB} to $q$, we get following result. 
\begin{theorem} 
	Let $\mathcal{A}$ be a commutative unital C*-algebra and $p(z)\coloneqq a_0+a_1z+\cdots +a_{n-1}z^{n-1}+z^n \in \mathcal{A}[z]$ with $a_0$ invertible.  If an invertible element $b \in \mathcal{A}$ satisfies $p(b)=0$, then 
	\begin{align*}
	\frac{1}{\|b^{-1}\|} \geq \frac{2}{2w_\mathcal{A}(R)+\sqrt{\left\|(a^{-1}_0)^*a^{-1}_0+\sum_{j=1}^{n-1}(a_ja_0^{-1})^*a_ja_0^{-1}\right\|^2}+\|a_{1}a_0^{-1}\|}.
	\end{align*}
\end{theorem}
Equality term in (\ref{FKRB}) comes from the numerical radius of right shift matrix obtained by Davidson and Holbrook \cite{DAVIDSONHOLBROOK} (also see \cite{HAAGERUPHARPE, KARAEV}). We are unable to do this for the right shift matrix over C*-algebras. 
We now derive Theorem \ref{DIT} for Hilbert C*-modules. 
\begin{theorem} (\textbf{Modular Dragomir Numerical Radius Inequality}) \label{MD} Let $\mathcal{E}$ be a  Hilbert C*-module over a  unital C*-algebra $\mathcal{A}$ and  $T:\mathcal{E} \to \mathcal{E} $ be a self-adjoint  morphism. Then
	\begin{align*}
		w_\mathcal{A}(T) \leq \frac{1}{\sqrt{2}}\sqrt{w_\mathcal{A}(T^2)+\|T\|^2}\leq  \frac{1}{\sqrt{2}}\sqrt{\|T^2\|+\|T\|^2}\leq \|T\|.
	\end{align*}
\end{theorem}
\begin{proof}
	Let $x\in  \mathcal{E}$ with $\langle x, x \rangle=1$. Then using Inequality (\ref{KI}) and the self-adjointness of $T$, we get 
		\begin{align*}
		\|\langle x, Tx\rangle\|^2&=\|\langle x, Tx\rangle\langle x, Tx\rangle^*\|=\|\langle x, Tx\rangle \langle  Tx, x\rangle\|\\
		&=\|\langle Tx, x\rangle \langle x, Tx\rangle\|\leq \frac{\|\langle Tx, Tx \rangle\|+\|Tx\|\|Tx\|}{2}\\
		&= \frac{\|\langle x, T^2x \rangle\|+\|Tx\|\|T^*x\|}{2}\leq  \frac{\|\langle x, T^2x \rangle\|+\|T\|\|x\|\|T\|\|x\|}{2}\\
		&=\frac{\|\langle x, T^2x \rangle\|+\|T\|^2}{2}\leq \frac{\sup_{y \in \mathcal{E}, \langle y, y \rangle =1}\|\langle y, T^2y\rangle\|+\|T\|^2}{2}\\
		&=\frac{w_\mathcal{A}(T^2)+\|T\|^2}{2}.
	\end{align*}
Therefore 
\begin{align*}
		w_\mathcal{A}(T)= \sup_{x \in \mathcal{E}, \langle x, x \rangle=1}\|\langle x, Tx\rangle\|\leq \frac{1}{\sqrt{2}}\sqrt{w_\mathcal{A}(T^2)+\|T\|^2}\leq  \frac{1}{\sqrt{2}}\sqrt{\|T^2\|+\|T\|^2}\leq \|T\|.
\end{align*}
\end{proof}
Note that  we derived Theorem \ref{MD} only for self-adjoint morphisms. We are unable to derive Theorem \ref{MD} for arbitrary adjointable morphisms. We also note that in the case of self-adjoint operator $T$ on Hilbert spaces, we have $w_\mathbb{C}(T)=\|T\|$ \cite{GUSTAFSONRAO} which we can't say for self-adjoint morphisms on Hilbert C*-modules.

\section{Noncommutative  Toeplitz-Hausdorff Problem}
Motivated from several breakthrough results in numerical range and radius, we formulate following problems. What is noncommutative 
\begin{enumerate}[\upshape(1)]
	\item 	 Toeplitz-Hausdorff Theorem (fundamental theorem of the numerical range) \cite{TOEPLITZ, GUTKIN, HAUSDORFF, MCINTOSH, DAVIS, RAGHAVENDRAN, GUSTAFSON, GUSTAFSONRAO, NARCOWICHWARD, STONE, FEINTUCHMARKUS, HALMOS}?
	\item  Wintner spectral inclusion theorem for numerical range \cite{BERBERIAN, GUSTAFSONRAO, WINTNER}?
	\item  Halmos-Bernau-Smithies-Berger power inequality \cite{PEARCY, BERNAUSMITHIES}?
	\item Alpin-Chien-Yeh inequality\cite{ALPINCHIENYEH}?
	\item  Kittaneh inequalities \cite{KITTANEH, KITTANEH2}?
	\item  Yamazaki inequality \cite{YAMAZAKI}?
	\item Kittaneh-Moslehian-Yamazaki inequality \cite{KITTANEHMOSLEHIANYAMAZAKI}?
	\item Abu-Omar-Kittaneh inequality \cite{ABUOMARKITTANEH}?
		\item   Bhunia-Bag-Paul  inequality \cite{BHUNIABAGPAUL}?
		\item  elliptical range theorem \cite{LI, BROWNSPITKOVSKY}?
	\item  Berger power dilation for numerical contraction \cite{GUSTAFSONRAO, BERGERSTAMPFLI, BERGERSTAMPFLI2, NAGYFOIASBOOK}?
	\item  Ando theorem for numerical contraction \cite{ANDO}?
	\item Furuta-Nakamoto theorem for numerical contractions \cite{FURUTANAKAMOTO2}?
		\item  El-Haddad-Kittaneh numerical radius inequality \cite{ELHADDADKITTANEH}?
			\item  description of numerical range of 3 $\times $ 3 matrices \cite{KEELERRODMANSPITKOVSKY}?
		\item  description of numerical range of 4 $\times $ 4 matrices \cite{GAU}?
			\item  Holbrook numerical radius inequalities for the product of commuting matrices \cite{HOLBROOK}?
		\item  Okubo-Ando numerical radius inequalities for the product of commuting matrices \cite{OKUBOANDO, MULLER, HOLBROOK2}?
		\item  Bouldin  numerical radius inequalities for the product of commuting matrices \cite{BOULDIN, BOULDIN2}?
		\item Poncelet property for numerical ranges \cite{ADAMSCORBETTGORKIN, WU, GAUWU2}?
		\item  Anderson theorem \cite{GAUWUBOOK, TAMYANG, DRITSCHELWOERDEMAN}?
		\item 	 Kippenhahn boundary generating curve theorem \cite{KIPPENHAHN, FIEDLER}?
		\item  Kippenhahn corner-point theorem \cite{KIPPENHAHN, FIEDLER}?
		\item   Hildebrandt normal eigenvalue theorem  \cite{BOURDONSHAPIRO, HILDEBRANDT}?
		\item  Hildebrandt intersection theorem \cite{WILLIAMS, FURUTANAKAMOTO}?
		\item  Marcus-Shure description for the numerical range of zero-one matrices \cite{MARCUSSHURE} and Davidson-Holbrook estimate for the numerical radius of zero-one matrices \cite{DAVIDSONHOLBROOK}?
		\item Johnson numerical range inclusion theorem \cite{JOHNSON2}?
	\item  Choi-Li constrained numerical range theorem \cite{CHOILI}?
	\item   Wang-Wu-Gau result for Crawford number \cite{WANGWUGAU}?
	\item  Abu-Omar-Kittaneh inequality (involving generalized Aluthge transform) \cite{ABUOMARKITTANEH2}?
	\item  Gau-Wu  numerical radius equality characterizations \cite{GAUWU3}?
	\item  Gau-Wu theorem for compact operators \cite{GAUWU4}?
	\item  Perron-Frobenius type results on the numerical range \cite{MAROULASPSARRAKOSTSATSOMEROS}?
	\item  Helton-Spitkovsky characterization of shapes of numerical ranges \cite{HELTONSPITKOVSKY}?
	\item  Radjabalipour-Radjavi result on  numerical ranges \cite{RADJABALIPOURRADJAVI}?
	\item  Pollack theorem on numerical ranges \cite{POLLACK}?
	\item  Anderson result on numerical ranges \cite{AGLER}?
	\item  Abu-Omar-Kittaneh numerical radius  inequalities for the product  of matrices \cite{ABUOMARKITTANEH3}?
	\item  Kittaneh-Moradi  inequality \cite{KITTANEHMORADI}?
		\item  Lancaster characterization for the closedness of numerical range \cite{LANCASTER}?
			\item  Gau-Wu characterization for numerical ranges of completely non-unitary contractions \cite{GAUWU5}?
		\item Mees-Atherton domains containing  numerical ranges \cite{MEESATHERTON}?
			\item   Chien-Tam characterizations of circularity of numerical ranges \cite{CHIENTAM}?
	\item  Shiu theorem on the growth of numerical range of powers of operator \cite{SHIU}?
	\end{enumerate}




\section{Noncommutative Crouzeix Inequality, von Neumann inequality and  Ando inequality Problems}
Breakthrough Crouzeix theorem says following. 
\begin{theorem} \cite{CROUZEIX, CROUZEIX2, CROUZEIX3, BICKELGORKIN, RANSFORDSCHWENNINGER} (\textbf{Crouzeix Theorem}) 
For every $d \in \mathbb{N}$ and for every matrix $M\in \mathbb{M}_{d}(\mathbb{C})$, we have 
\begin{align*}
(\textbf{Crouzeix-Palencia Inequality})	\quad \quad \quad	\|p(M)\|\leq (1+\sqrt{2}) \sup\left\{|p(z)|: z \in W_\mathbb{C}(M)\right\},  \quad \forall p \in \mathbb{C}[z].
\end{align*}	
\end{theorem}

We formulate following problem. 
\begin{problem} (\textbf{Noncommutative Crouzeix Problem})
	Let $\mathcal{A}$ be a unital C*-algebra.  Whether there is a universal constant $R_\mathcal{A}$ (which may depend upon $\mathcal{A}$) satisfying following: For every $d \in \mathbb{N}$ and for every matrix $M\in \mathbb{M}_{d}(\mathcal{A})$, we have 
	\begin{align*}
		\|p(M)\|\leq R_\mathcal{A} \sup\left\{\|p(z)\|: z \in W_\mathcal{A}(M)\right\},  \quad \forall p \in \mathcal{A}[z].
	\end{align*}
\end{problem}
An inequality which is very close to Crouzeix inequality is the von Neumann inequality. 
\begin{theorem} \label{VT}
	\cite{VONNEUMANN, NAGY, NAGYFOIASBOOK, DELYONDELYON, NELSON} (\textbf{von Neumann Theorem}) 
	For every $d \in \mathbb{N}$ and for every matrix $M\in \mathbb{M}_{d}(\mathbb{C})$ with $\|M\|\leq 1$,  we have 
	\begin{align*}
		(\textbf{von Neumann Inequality})	\quad \quad \quad 	\|p(M)\|\leq  \sup\left\{|p(z)|: z \in \mathbb{C}, |z|\leq 1\right\},  \quad \forall p \in \mathbb{C}[z].
	\end{align*}	
\end{theorem}
Based on Theorem \ref{VT}  we formulate following problems. 
\begin{problem} (\textbf{Noncommutative von Neumann Inequality Problem}) 
	Let $\mathcal{A}$ be a unital C*-algebra. Whether there is a universal constant $R_\mathcal{A}$ (which may depend upon $\mathcal{A}$) satisfying following: For every $d \in \mathbb{N}$ and for every $A\in \mathbb{M}_{d}(\mathcal{A})$ with $\|A\|\leq 1$, we have 
	\begin{align*}
		\|p(A)\|\leq R_\mathcal{A} \sup\left\{\|p(z)\|: z \in \mathcal{A}, \|z\|\leq 1\right\},  \quad \forall p \in \mathcal{A}[z].
	\end{align*}
\end{problem}
Theorem \ref{VT} has been extended by Ando for two variables. 
\begin{theorem} \label{AT}
	\cite{ANDO, NAGYFOIASBOOK} (\textbf{Ando Theorem}) 
	For every $d \in \mathbb{N}$ and for all matrices $M, N\in \mathbb{M}_{d}(\mathbb{C})$ with $\|M\|\leq 1$, $\|N\|\leq 1$ and $MN=NM$,  we have 
	\begin{align*}
		(\textbf{Ando Inequality})	\quad \quad \quad 	\|p(M, N)\|\leq  \sup\left\{|p(z, w)|: z, w \in \mathbb{C}, |z|\leq 1, |w|\leq 1\right\},  \quad \forall p \in \mathbb{C}[z, w].
	\end{align*}		
\end{theorem}
Based on Theorem \ref{AT}, we formulate following problem. 
\begin{problem}
	(\textbf{Noncommutative Ando Inequality Problem}) 
	Let $\mathcal{A}$ be a unital C*-algebra. Whether there is a universal constant $R_\mathcal{A}$ (which may depend upon $\mathcal{A}$) satisfying following: For every $d \in \mathbb{N}$ and for all $A, B\in \mathbb{M}_{d}(\mathcal{A})$ with $\|A\|\leq 1$, $\|B\|\leq 1$ and $AB=BA$, we have
	\begin{align*}
		\quad \quad \quad 	\|p(A, B)\|\leq  \sup\left\{\|p(z, w)\|: z, w \in \mathcal{A}, \|z\|\leq 1, \|w\|\leq 1\right\},  \quad \forall p \in \mathcal{A}[z, w].
	\end{align*}		
\end{problem}
Note that Ando theorem cannot be extended to more than two commuting matrices \cite{VAROPOULOS, CRABBDAVIE}. We observe that the Halmos dilation \cite{HALMOS2}, Egervary dilation \cite{LEVYSHALIT, EGERVARY} and  Sz.-Nagy dilation (Schaffer construction) \cite{SCHAFFER} carry over to adjointable morphisms on Hilbert C*-modules. This observation will give following result. 
\begin{theorem} (\textbf{Noncommutative Sz.-Nagy Dilation})\label{NS}
	Let $\mathcal{E}$ be a Hilbert C*-module over a unital  C*-algebra $\mathcal{A}$. Let 	$T :\mathcal{E}\to \mathcal{E}$ be an adjointable morphism such that $\|T\|\leq1$. Then there exist a Hilbert C*-module $\mathcal{E}_0$ which contains $\mathcal{E}$ isometrically, $\mathcal{E}$ is orthogonally complementable in $\mathcal{E}_0$,  $P_\mathcal{E}:\mathcal{E}_0\to \mathcal{E}$ is an onto orthogonal projection and unitary  morphism $U:\mathcal{E}_0\to \mathcal{E}_0$ such that 
	\begin{align*}
		T^nx=P_\mathcal{E} U^nx, \quad \forall n \in \mathbb{N}, \forall x \in \mathcal{E}.
	\end{align*}
\end{theorem}

\begin{corollary}
	Let $\mathcal{E}$ be a Hilbert C*-module over a unital  C*-algebra $\mathcal{A}$. Let 	$T :\mathcal{E}\to \mathcal{E}$ be an adjointable morphism such that $\|T\|\leq1$. Let $\mathcal{E}_0$, $P_\mathcal{E}$ and $U$ be as in Theorem \ref{NS}. Then 
	\begin{align*}
		\|p(T)\|\leq \|p(U)\|, \quad \forall p(z)=a_0+za_1+\cdots+z^n a_n \in \mathcal{A}[z].
	\end{align*}
\end{corollary}
\begin{corollary}
	Let $\mathcal{E}$ be a Hilbert C*-module over a unital  C*-algebra $\mathcal{A}$. Let 	$T :\mathcal{E}\to \mathcal{E}$ be an adjointable morphism such that $\|T\|\leq1$. Let $x \in \mathcal{E}$ be such that $Tx=x$. Then $T^*x=x$.
\end{corollary}
\begin{proof}
	Let $x \in \mathcal{E}$ be such that $Tx=x$.	Let $\mathcal{E}_0$, $P_\mathcal{E}$ and $U$ be as in Theorem \ref{NS}. We then have 
	\begin{align*}
		x=Tx=P_\mathcal{E}Ux, \quad \langle Ux, Ux \rangle =\langle x, x \rangle, \quad P_\mathcal{E}x=x. 
	\end{align*}
	We see that 
	\begin{align*}
		\langle Ux-x, Ux-x\rangle &=2\langle x, x \rangle -\langle Ux, x \rangle-\langle x, Ux \rangle\\
		&=2\langle x, x \rangle -\langle Ux, P_\mathcal{E}x \rangle-\langle P_\mathcal{E}x, Ux \rangle\\
		&=2\langle x, x \rangle -\langle P_\mathcal{E}Ux, x \rangle-\langle x, P_\mathcal{E}Ux \rangle\\
		&=2\langle x, x \rangle -\langle Tx, x \rangle-\langle x, Tx \rangle\\
		&=2\langle x, x \rangle -\langle x, x \rangle-\langle x, x \rangle=0.
	\end{align*}
	Therefore 
	\begin{align*}
		Ux=x \implies x=U^*x.
	\end{align*}
	Hence 
	\begin{align*}
		x=U^*x=P_\mathcal{E}U^*x=T^*x.
	\end{align*}
\end{proof}
We also observe that the proof of Ando dilation \cite{ANDO} for commuting operators on Hilbert space will not carry over to commuting adjointable morphisms on Hilbert C*-modules (mainly because submodules need not be orthogonally complementable and C*-algebras need not have invariant basis number property). 

\section{Acknowledgments}
AI is used for the literature survey and grammar improvement.


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