Noncommutative Buzano-Dragomir Inequality and Applications

1. Introduction

More than half-century ago, Buzano derived the following generalization of the Cauchy-Schwarz inequality [1].

Theorem 1. 

[1,2,3] (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

$$\begin{array}{c}\hfill \left|\langle \tau ,h\rangle \langle h,\omega \rangle \right|\le {\displaystyle \frac{{\parallel h\parallel }^2\left(\right|\langle \tau ,\omega \rangle |+\parallel \tau \parallel \parallel \omega \parallel )}{2}}\le {\parallel h\parallel }^2\parallel \tau \parallel \parallel \omega \parallel ,\forall h,\tau ,\omega \in \mathcal{H}.\end{array}$$

(1)

In other words,

$$\begin{array}{c}\hfill \left|〈\tau ,{\displaystyle \frac{h}{\parallel h\parallel }}〉〈{\displaystyle \frac{h}{\parallel h\parallel }},\omega 〉\right|\le {\displaystyle \frac{|\langle \tau ,\omega \rangle |+\parallel \tau \parallel \parallel \omega \parallel }{2}}\le \parallel \tau \parallel \parallel \omega \parallel ,\forall \tau ,\omega \in \mathcal{H},\forall h\in \mathcal{H}\setminus \left\{0\right\}.\end{array}$$

Further, we have following.

(i)

Let $\tau ,\omega \in \mathcal{H}\setminus \left\{0\right\}$ be such that $\langle \tau ,\omega \rangle =0$. Choose $\alpha ,\beta \in \mathbb{C}$ with $\left|\beta \right|=1$. Define

$$\begin{array}{c}\hfill h_{\alpha ,\beta }\alpha \left({\displaystyle \frac{\tau }{\parallel \tau \parallel }}+\beta {\displaystyle \frac{\omega }{\parallel \omega \parallel }}\right).\end{array}$$

Then

$$\begin{array}{c}\hfill |\langle \tau ,h_{\alpha ,\beta }\rangle \langle h_{\alpha ,\beta },\omega \rangle |={\displaystyle \frac{\parallel h_{\alpha ,\beta }{\parallel }^2\left(\right|\langle \tau ,\omega \rangle |+\parallel \tau \parallel \parallel \omega \parallel )}{2}}.\end{array}$$

(ii)

Let $\tau ,\omega \in \mathcal{H}\setminus \left\{0\right\}$ be such that $\langle \tau ,\omega \rangle \ne 0$. For $\alpha \in \mathbb{C}$, define

$$\begin{array}{c}\hfill h_{\alpha }\alpha \left({\displaystyle \frac{\tau }{\parallel \tau \parallel }}+{\displaystyle \frac{\langle \tau ,\omega \rangle }{|\langle \tau ,\omega \rangle |}}{\displaystyle \frac{\omega }{\parallel \omega \parallel }}\right).\end{array}$$

Then

$$\begin{array}{c}\hfill |\langle \tau ,h_{\alpha }\rangle \langle h_{\alpha },\omega \rangle |={\displaystyle \frac{\parallel h_{\alpha }{\parallel }^2\left(\right|\langle \tau ,\omega \rangle |+\parallel \tau \parallel \parallel \omega \parallel )}{2}}.\end{array}$$

Ten years ago, Dragomir made the following far reaching generalization of Theorem 1 [4].

Theorem 2. 

[4] (Buzano-Dragomir Inequality) Let $\mathcal{H}$ be a complex Hilbert space and $P:\mathcal{H}\to \mathcal{H}$ be an orthogonal projection. Then

$$\begin{array}{c}\hfill \left|\langle P\tau ,\omega \rangle \right|\le {\displaystyle \frac{|\langle \tau ,\omega \rangle |+\parallel \tau \parallel \parallel \omega \parallel }{2}},\forall \tau ,\omega \in \mathcal{H}.\end{array}$$

(2)

Now note that for any nonzero $h\in \mathcal{H}$, the map

$$\begin{array}{c}\hfill \mathcal{H}\ni \tau \mapsto 〈\tau ,{\displaystyle \frac{h}{\parallel h\parallel }}〉{\displaystyle \frac{h}{\parallel h\parallel }}\in \mathcal{H}\end{array}$$

is a rank one orthogonal projection. Thus Inequality (2) recovers Inequality (1). Inequality (1) has various applications, we describe few. Recall that [5,6,7,8,9] the numerical range of a bounded linear operator $T:\mathcal{H}\to \mathcal{H}$ is defined as

$$\begin{array}{c}\hfill W_{\mathbb{C}}\left(T\right)\{\langle Th,h\rangle :h\in \mathcal{H},\parallel h\parallel =1\}\end{array}$$

and the numerical radius of T is defined as

$$\begin{array}{c}\hfill w_{\mathbb{C}}\left(T\right)\underset{h\in \mathcal{H},\parallel h\parallel =1}{sup}\left|\langle Th,h\rangle \right|.\end{array}$$

For $\tau ,\omega \in \mathcal{H}$, define the rank one operator

$$\begin{array}{c}\hfill \tau \otimes \omega :\mathcal{H}\ni h\mapsto (\tau \otimes \omega )h\langle h,\omega \rangle \tau \in \mathcal{H}.\end{array}$$

In 1993, Fujii and Kubo derived following result, using Inequality (1) [2].

Theorem 3. 

[2] Let $\mathcal{H}$ be a complex Hilbert space and $\tau ,\omega \in \mathcal{H}$. Then

$$\begin{array}{c}\hfill w_{\mathbb{C}}(\tau \otimes \omega )={\displaystyle \frac{|\langle \tau ,\omega \rangle |+\parallel \tau \parallel \parallel \omega \parallel }{2}}.\end{array}$$

Let $p\left(z\right)a_0+a_{1}z+\dots +a_{n-1}{z}^{n-1}+z^n\in \mathbb{C}\left[z\right]$. A direct observation reveals that the zeros of p are the eigenvalues of the Frobenius companion matrix

$$\begin{array}{c}\hfill C_p\left(\begin{array}{ccccccc}-a_{n-1}& -a_{n-2}& -a_{n-3}& \dots & -a_2& -a_1& -a_0\\ 1& 0& 0& \dots & 0& 0& 0\\ 0& 1& 0& \dots & 0& 0& 0\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & 0& 0& 0\\ 0& 0& 0& \dots & 1& 0& 0\\ 0& 0& 0& \dots & 0& 1& 0\end{array}\right)\in {\mathbb{M}}_n\left(\mathbb{C}\right)\end{array}$$

and the eigenvalues of $C_p$ are the same as the zeros of p. Let R be the right shift matrix defined by

$$\begin{array}{c}\hfill R\left(\begin{array}{ccccccc}0& 0& 0& \dots & 0& 0& 0\\ 1& 0& 0& \dots & 0& 0& 0\\ 0& 1& 0& \dots & 0& 0& 0\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & 0& 0& 0\\ 0& 0& 0& \dots & 1& 0& 0\\ 0& 0& 0& \dots & 0& 1& 0\end{array}\right)\in {\mathbb{M}}_n\left(\mathbb{C}\right).\end{array}$$

Define

$$\begin{array}{c}\hfill \mathbf{a}\left(\begin{array}{c}\overline{a_{n-1}}\\ \overline{a_{n-2}}\\ \overline{a_{n-3}}\\ ⋮\\ \overline{a_2}\\ \overline{a_1}\\ \overline{a_0}\end{array}\right),{\mathbf{e}}_{\mathbf{1}}\left(\begin{array}{c}1\\ 0\\ 0\\ ⋮\\ 0\\ 0\\ 0\end{array}\right).\end{array}$$

Then

$$\begin{array}{c}\hfill C_p=R-{\mathbf{e}}_{\mathbf{1}}\otimes \mathbf{a}.\end{array}$$

(3)

Fujii and Kubo derived following bound for the zeros of p using Inequality 1 and the representation (3) [2].

Theorem 4. 

[2] (Fujii-Kubo polynomial root upper bound) Let $p\left(z\right)a_0+a_{1}z+\dots +a_{n-1}{z}^{n-1}+z^n\in \mathbb{C}\left[z\right]$. If λ is a zero of p, then

$$\begin{array}{cc}\hfill \left|\lambda \right|& \le w_{\mathbb{C}}\left(R\right)+{\displaystyle \frac{\sqrt{{\sum }_{j=0}^{n-1}{\left|a_j\right|}^2}+\left|a_{n-1}\right|}{2}}=cos\left({\displaystyle \frac{\pi }{n+1}}\right)+{\displaystyle \frac{\sqrt{{\sum }_{j=0}^{n-1}{\left|a_j\right|}^2}+\left|a_{n-1}\right|}{2}}.\hfill \end{array}$$

(4)

Let $p\left(z\right)a_0+a_{1}z+\dots +a_{n-1}{z}^{n-1}+z^n\in \mathbb{C}\left[z\right]$ with $a_0\ne 0$. Define

$$\begin{array}{c}\hfill q\left(z\right){\displaystyle \frac{1}{a_0}}{z}^{n}p\left({\displaystyle \frac{1}{z}}\right)={\displaystyle \frac{1}{a_0}}+{\displaystyle \frac{a_{n-1}}{a_0}}z+\dots +{\displaystyle \frac{a_1}{a_0}}{z}^{n-1}+z^n\in \mathbb{C}\left[z\right].\end{array}$$

We note that $\lambda \in \mathbb{C}\setminus \left\{0\right\}$ satisfies $p\left(\lambda \right)=0$ if and only if $q(1/\lambda )=0$ [7]. Frobenius companion matrix of q is

$$\begin{array}{c}\hfill C_q\left(\begin{array}{ccccccc}-{\displaystyle \frac{a_1}{a_0}}& -{\displaystyle \frac{a_2}{a_0}}& -{\displaystyle \frac{a_3}{a_0}}& \dots & -{\displaystyle \frac{a_{n-1}}{a_0}}& -{\displaystyle \frac{a_{n-1}}{a_0}}& -{\displaystyle \frac{1}{a_0}}\\ 1& 0& 0& \dots & 0& 0& 0\\ 0& 1& 0& \dots & 0& 0& 0\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & 0& 0& 0\\ 0& 0& 0& \dots & 1& 0& 0\\ 0& 0& 0& \dots & 0& 1& 0\end{array}\right)\in {\mathbb{M}}_n\left(\mathbb{C}\right)\end{array}$$

By applying Theorem 4 to q, we get following result.

Theorem 5. 

[2] (Fujii-Kubo polynomial root lower bound) Let $p\left(z\right)a_0+a_{1}z+\dots +a_{n-1}{z}^{n-1}+z^n\in \mathbb{C}\left[z\right]$ with $a_0\ne 0$. If λ is a zero of p, then

$$\begin{array}{c}\hfill \left|\lambda \right|\ge {\displaystyle \frac{2|a_0|}{2|a_0|cos\left(\frac{\pi }{n+1}\right)+\sqrt{1+{\sum }_{j=1}^{n-1}{\left|a_j\right|}^2}+\left|a_1\right|}}.\end{array}$$

For any bounded linear operator $T:\mathcal{H}\to \mathcal{H}$, the Cauchy-Schwarz inequality gives

$$\begin{array}{c}\hfill w_{\mathbb{C}}\left(T\right)\le \parallel T\parallel .\end{array}$$

(5)

In 2007, Dragomir strengthened Inequality (5) using Buzano inequality [10].

Theorem 6. 

[10] (Dragomir Numerical Radius Inequality) Let $T:\mathcal{H}\to \mathcal{H}$ be a bounded linear operator. Then

$$\begin{array}{c}\hfill w_{\mathbb{C}}\left(T\right)\le {\displaystyle \frac{1}{\sqrt{2}}}\sqrt{w_{\mathbb{C}}\left(T^2\right)+{\parallel T\parallel }^2}\le \parallel T\parallel .\end{array}$$

Our fundamental motivation comes from the following question: What is the noncommutative analogue of Theorem 2? This is then naturally connected with the notion of Hilbert C*-modules which are first introduced by Kaplansky [11] for modules over commutative C*-algebras and later developed for modules over arbitrary C*-algebras by Paschke [12] and Rieffel [13].

Definition 1. 

[11,12,13] Let $\mathcal{A}$ be a unital C*-algebra. A right module $\mathcal{E}$ over $\mathcal{A}$ is said to be asemi-inner product C*-moduleif there exists a map $\langle ·,·\rangle :\mathcal{E}\times \mathcal{E}\to \mathcal{A}$ such that the following hold.

(i)

$\langle x,x\rangle \ge 0$, $\forall x\in \mathcal{E}$.

(ii)

$\langle x+y,z\rangle =\langle x,z\rangle +\langle y,z\rangle$, $\forall x,y,z\in \mathcal{E}$.

(iii)

$\langle x,ya\rangle =\langle x,y\rangle a$, $\forall x,y\in \mathcal{E}$, $\forall a\in \mathcal{A}$.

(iv)

$\langle x,y\rangle ={\langle y,x\rangle }^{*}$, $\forall x,y\in \mathcal{E}$.

Semi-inner product C*-modules satisfy noncommutative Cauchy-Schwarz inequality.

Theorem 7. 

[14] (Kaplansky-Paschke-Rieffel Inequality) Let $\mathcal{E}$ be a semi-inner product C*-module. Then

$$\begin{array}{c}\hfill \langle y,x\rangle \langle x,y\rangle \le \parallel \langle x,x\rangle \parallel \langle y,y\rangle ,\forall x,y\in \mathcal{E}.\end{array}$$

In particular,

$$\begin{array}{c}\hfill {\parallel \langle y,x\rangle \parallel }^2\le \parallel \langle x,x\rangle \parallel \parallel \langle y,y\rangle \parallel ,\forall x,y\in \mathcal{E}.\end{array}$$

Definition 2. 

[14] A semi-inner product C*-module $\mathcal{E}$ is said to be aninner product C*-moduleif $x\in \mathcal{E}$ satisfies $\langle x,x\rangle =0$, then $x=0$.

Definition 3. 

[14] A inner product C*-module $\mathcal{E}$ is said to beHilbert C*-moduleif $\mathcal{E}$ is complete w.r.t. the norm $\parallel x\parallel \sqrt{\parallel \langle x,x\rangle \parallel }$, $\forall x\in \mathcal{E}$.

In 2012, Khosravi, Drnovsek and Moslehian derived noncommutative version of Inequality (1) [15].

Theorem 8. 

[15] (Buzano-Khosravi-Drnovsek-Moslehian Inequality) Let $\mathcal{E}$ be a Hilbert C*-module. Then

$$\begin{array}{c}\hfill \parallel \langle \omega ,x\rangle \langle x,\tau \rangle \parallel \le {\displaystyle \frac{\parallel \langle \tau ,\omega \rangle \parallel +\parallel \tau \parallel \parallel \omega \parallel }{2}}\le \parallel \tau \parallel \parallel \omega \parallel ,\forall \tau ,\omega \in \mathcal{E},\forall x\in \mathcal{E}\mathrm{with}\langle x,x\rangle =1.\end{array}$$

(6)

In this article, we show that Theorem 2 extends to orthogonal projections on Hilbert C*-modules (which will also generalize Theorem 8). We also derive modular analogue of Theorems 4 and 6.

2. Noncommutative Buzano-Dragomir Inequality

We start by deriving noncommutative analogue of Theorem 2.

Theorem 9.(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

$$\begin{array}{c}\hfill \parallel \langle \tau ,P\omega \rangle \parallel \le {\displaystyle \frac{\parallel \langle \tau ,\omega \rangle \parallel +\parallel \tau \parallel \parallel \omega \parallel }{2}},\forall \tau ,\omega \in \mathcal{E}.\end{array}$$

Proof. 

We find

$$\begin{array}{cc}\hfill \parallel 2\langle \tau ,P\omega \rangle -\langle \tau ,\omega \rangle \parallel & =\parallel \langle \tau ,2P\omega -\omega \rangle \parallel \le \parallel \tau \parallel \parallel 2P\omega -\omega \parallel .\hfill \end{array}$$

We now find

$$\begin{array}{cc}\hfill {\parallel 2P\omega -\omega \parallel }^2& =\parallel \langle 2P\omega -\omega ,2P\omega -\omega \rangle \parallel \hfill \\ \hfill & =\parallel 4\langle P\omega ,P\omega \rangle -2\langle P\omega ,\omega \rangle -2\langle \omega ,P\omega \rangle +\langle \omega ,\omega \rangle \parallel \hfill \\ \hfill & =\parallel \langle \omega ,\omega \rangle \parallel \hfill \\ \hfill & ={\parallel \omega \parallel }^2.\hfill \end{array}$$

Therefore

$$\begin{array}{cc}\hfill 2\parallel \langle \tau ,P\omega \rangle \parallel -\parallel \langle \tau ,\omega \rangle \parallel & \le \parallel 2\langle \tau ,P\omega \rangle -\langle \tau ,\omega \rangle \parallel \le \parallel \tau \parallel \parallel \omega \parallel .\hfill \end{array}$$

□

Corollary 1. 

Theorem 8 follows from Theorem 9.

Proof. 

Let $\mathcal{E}$ be a Hilbert C*-module and $x\in \mathcal{E}$ with $\langle x,x\rangle =1$. Define

$$\begin{array}{c}\hfill P:\mathcal{E}\ni y\mapsto Pyx\langle x,y\rangle \in \mathcal{E}.\end{array}$$

Then P is an orthogonal projection. Theorem 9 gives

$$\begin{array}{cc}\hfill \parallel \langle \tau ,x\rangle \langle x,\omega \rangle \parallel & =\parallel \langle \tau ,x\langle x,\omega \rangle \rangle \parallel =\parallel \langle \tau ,P\omega \rangle \parallel \hfill \\ \hfill & \le {\displaystyle \frac{\parallel \langle \tau ,\omega \rangle \parallel +\parallel \tau \parallel \parallel \omega \parallel }{2}},\forall \tau ,\omega \in \mathcal{E}.\hfill \end{array}$$

□

In 2016, Dragomir further improved Theorem 2 for bounded linear operators [16].

Theorem 10. 

[16] (Buzano-Dragomir Inequality) Let $\mathcal{H}$ be a complex Hilbert space and $T:\mathcal{H}\to \mathcal{H}$ be a bounded linear operator. Then

$$\begin{array}{c}\hfill \left|\langle T\tau ,T\omega \rangle \right|\le {\displaystyle \frac{{\parallel T\parallel }^2\left(\right|\langle \tau ,\omega \rangle |+\parallel \tau \parallel \parallel \omega \parallel )}{2}},\forall \tau ,\omega \in \mathcal{H}.\end{array}$$

We now derive following noncommutative version of Theorem 10.

Theorem 11.(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

$$\begin{array}{c}\hfill \parallel \langle T\tau ,T\omega \rangle \parallel \le {\parallel T\parallel }^2\parallel \langle \tau ,\omega \rangle \parallel +{\parallel \langle \tau ,\tau \rangle \parallel T\parallel }^2-\langle T\tau ,T\tau \rangle {\parallel }^{{\displaystyle \frac{1}{2}}}\parallel \langle \omega ,\omega \rangle {\parallel T\parallel }^2-\langle T\omega ,T\omega \rangle {\parallel }^{{\displaystyle \frac{1}{2}}},\forall \tau ,\omega \in \mathcal{E}.\end{array}$$

Proof. 

Our proof is motivated by the Hilbert space argument due to Dragomir [16]. Let $T:\mathcal{E}\to \mathcal{E}$ be an adjointable morphism and $\tau ,\omega \in \mathcal{E}$. Define

$$\begin{array}{c}\hfill [·,·]:\mathcal{E}\times \mathcal{E}\ni (x,y)\mapsto {[x,y]\langle x,y\rangle \parallel T\parallel }^2-\langle Tx,Ty\rangle \in \mathcal{A}.\end{array}$$

Then $[·,·]$ is a semi-inner product. By applying Theorem 7 to this semi-inner product, we get

$$\begin{array}{c}\hfill \parallel [x,y]{\parallel }^2\le \parallel [x,x]\parallel \parallel [y,y]\parallel ,\forall x,y\in \mathcal{E}.\end{array}$$

In particular,

$$\begin{array}{c}\hfill {∥[\tau ,\omega ]∥}^2\le \parallel [\tau ,\tau ]\parallel \parallel [\omega ,\omega ]\parallel .\end{array}$$

Previous inequality gives

$$\begin{array}{c}\hfill {\parallel \langle \tau ,\omega \rangle \parallel T\parallel }^2-\langle T\tau ,T\omega \rangle {\parallel }^2\le {\parallel \langle \tau ,\tau \rangle \parallel T\parallel }^2-\langle T\tau ,T\tau \rangle \parallel \parallel \langle \omega ,\omega \rangle {\parallel T\parallel }^2-\langle T\omega ,T\omega \rangle \parallel .\end{array}$$

Therefore

$$\begin{array}{cc}\hfill \parallel \langle T\tau ,T\omega \rangle \parallel -{\parallel T\parallel }^2\parallel \langle \tau ,\omega \rangle \parallel & \le {\parallel \langle \tau ,\omega \rangle \parallel T\parallel }^2-\langle T\tau ,T\omega \rangle \parallel \hfill \\ \hfill & \le {\parallel \langle \tau ,\tau \rangle \parallel T\parallel }^2-\langle T\tau ,T\tau \rangle {\parallel }^{{\displaystyle \frac{1}{2}}}\parallel \langle \omega ,\omega \rangle {\parallel T\parallel }^2-\langle T\omega ,T\omega \rangle {\parallel }^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

□

We are unable to simplify Theorem 11. We therefore ask following problem.

Problem 12. 

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

$$\begin{array}{c}\hfill \parallel \langle T\tau ,T\omega \rangle \parallel \le {\displaystyle \frac{{\parallel T\parallel }^2(\parallel \langle \tau ,\omega \rangle \parallel +\parallel \tau \parallel \parallel \omega \parallel )}{2}},\forall \tau ,\omega \in \mathcal{E}?\end{array}$$

Let $\mathcal{E}$ be a Hilbert C*-module over a unital C*-algebra $\mathcal{A}$. Similar to Hilbert space case, we define the noncommutative numerical range of an adjointable morphism $T:\mathcal{E}\to \mathcal{E}$ is defined as

$$\begin{array}{c}\hfill W_{\mathcal{A}}\left(T\right)\{\langle x,Tx\rangle :x\in \mathcal{E},\langle x,x\rangle =1\}\subseteq \mathcal{A}\end{array}$$

and the noncommutative numerical radius T is defined as

$$\begin{array}{c}\hfill w_{\mathcal{A}}\left(T\right)\underset{x\in \mathcal{E},\langle x,x\rangle =1}{sup}\parallel \langle x,Tx\rangle \parallel \ge 0.\end{array}$$

Note that our definition of numerical range and radius on Hilbert C*-modules differ from the existing same notion on Hilbert C*-modules [17,18,19] We give various examples. In all following examples, $\mathcal{A}$ is a unital C*-algebra.

Example 1. 

Let $b\in \mathcal{A}$. Then

$$\begin{array}{c}\hfill W_{\mathcal{A}}\left(b\right)=\{a^{*}ba:a\in \mathcal{A},a^{*}a=1\}\end{array}$$

and

$$\begin{array}{cc}\hfill w_{\mathcal{A}}\left(b\right)& =sup\{\parallel a^{*}ba\parallel :a\in \mathcal{A},a^{*}a=1\}\hfill \\ \hfill & =\parallel b\parallel .\hfill \end{array}$$

If $\mathcal{A}$ is commutative, then $W_{\mathcal{A}}\left(b\right)=\left\{b\right\}$.

Example 2. 

Let $d\in \mathbb{N}$ and $I_d$ be the identity matrix of size d by d. Then we have

$$\begin{array}{c}\hfill W_{\mathcal{A}}\left(I_d\right)=\{\langle x,x\rangle :x\in {\mathcal{A}}^d,\langle x,x\rangle =1\}=\left\{1\right\}\end{array}$$

and

$$\begin{array}{c}\hfill w_{\mathcal{A}}\left(I_d\right)=1.\end{array}$$

Example 3. 

Define

$$\begin{array}{c}\hfill M\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right).\end{array}$$

Then

$$\begin{array}{cc}\hfill W_{\mathcal{A}}\left(M\right)& =\{\langle x,Mx\rangle :x\in {\mathcal{A}}^2,\langle x,x\rangle =1\}\hfill \\ \hfill & =\left\{a^{*}b:a,b\in \mathcal{A},a^{*}a+b^{*}b=1\right\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill w_{\mathcal{A}}\left(M\right)& =sup\left\{\parallel a^{*}b\parallel :a,b\in \mathcal{A},a^{*}a+b^{*}b=1\right\}.\hfill \end{array}$$

We note that

$$\begin{array}{c}\hfill a^{*}b{b}^{*}a\le {\displaystyle \frac{a^{*}a+b^{*}b}{2}},\forall a,b\in \mathcal{A}.\end{array}$$

Therefore

$$\begin{array}{cc}\hfill w_{\mathcal{A}}\left(M\right)& =sup\left\{\parallel a^{*}b\parallel :a,b\in \mathcal{A},a^{*}a+b^{*}b=1\right\}\hfill \\ \hfill & \le {\displaystyle \frac{1}{\sqrt{2}}}sup\{\sqrt{\parallel a^{*}a+b^{*}b\parallel }:a,b\in \mathcal{A},a^{*}a+b^{*}b=1\}\hfill \\ \hfill & ={\displaystyle \frac{1}{\sqrt{2}}}.\hfill \end{array}$$

Example 4. 

Let $d\in \mathbb{N}$ and $b_1,\dots ,b_d\in \mathcal{A}$. Define $diag\left({\left(b_j\right)}_{j=1}^d\right)$ as the diagonal matrix with diagonal entries $b_1,\dots ,b_d$. Then

$$\begin{array}{c}\hfill W_{\mathcal{A}}(diag\left({\left(b_j\right)}_{j=1}^d\right))=\left\{{\displaystyle \sum _{j=1}^d}{a}_j^{*}{b}_j{a}_j:{\left(a_j\right)}_{j=1}^d\in {\mathcal{A}}^d,{\displaystyle \sum _{j=1}^d}{a}_j^{*}{a}_j=1\right\}\end{array}$$

and

$$\begin{array}{cc}\hfill w_{\mathcal{A}}(diag\left({\left(b_j\right)}_{j=1}^d\right))& =sup\left\{∥{\displaystyle \sum _{j=1}^d}{a}_j^{*}{b}_j{a}_j∥:{\left(a_j\right)}_{j=1}^d\in {\mathcal{A}}^d,{\displaystyle \sum _{j=1}^d}{a}_j^{*}{a}_j=1\right\}\hfill \\ \hfill & \le {\displaystyle \sum _{j=1}^d}\parallel b_j\parallel .\hfill \end{array}$$

The set $W_{\mathcal{A}}(diag\left({\left(b_j\right)}_{j=1}^d\right))$ is the C*-convex hull of ${\left\{b_j\right\}}_{j=1}^d$ [20].

Example 5. 

Let $R\in {\mathbb{M}}_d\left(\mathcal{A}\right)$ be the right shift matrix defined by

$$\begin{array}{c}\hfill R\left(\begin{array}{ccccccc}0& 0& 0& \dots & 0& 0& 0\\ 1& 0& 0& \dots & 0& 0& 0\\ 0& 1& 0& \dots & 0& 0& 0\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & 0& 0& 0\\ 0& 0& 0& \dots & 1& 0& 0\\ 0& 0& 0& \dots & 0& 1& 0\end{array}\right).\end{array}$$

Thus the action of R is

$$\begin{array}{c}\hfill R{\left(a_j\right)}_{j=1}^d=(0,a_1,a_2,\dots ,a_{d-1}),\forall {\left(a_j\right)}_{j=1}^d\in {\mathcal{A}}^d.\end{array}$$

We then have

$$\begin{array}{c}\hfill \langle {\left(a_j\right)}_{j=1}^d,R{\left(a_j\right)}_{j=1}^d\rangle =a_2^{*}{a}_1+\dots +a_d^{*}{a}_{d-1},\forall {\left(a_j\right)}_{j=1}^d\in {\mathcal{A}}^d.\end{array}$$

Hence

$$\begin{array}{cc}\hfill \parallel \langle {\left(a_j\right)}_{j=1}^d,R{\left(a_j\right)}_{j=1}^d\rangle \parallel & =\parallel a_2^{*}{a}_1+\dots +a_d^{*}{a}_{d-1}\parallel \hfill \\ \hfill & \le {∥{\displaystyle \sum _{j=2}^d}{a}_j^{*}{a}_j∥}^{{\displaystyle \frac{1}{2}}}{∥{\displaystyle \sum _{k=1}^{d-1}}{a}_k^{*}{a}_k∥}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le {∥{\displaystyle \sum _{j=1}^d}{a}_j^{*}{a}_j∥}^{{\displaystyle \frac{1}{2}}}{∥{\displaystyle \sum _{k=1}^d}{a}_k^{*}{a}_k∥}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & =∥{\displaystyle \sum _{j=1}^d}{a}_j^{*}{a}_j∥,\forall {\left(a_j\right)}_{j=1}^d\in {\mathcal{A}}^d.\hfill \end{array}$$

Therefore

$$\begin{array}{c}\hfill W_{\mathcal{A}}\left(R\right)=\left\{a_2^{*}{a}_1+\dots +a_d^{*}{a}_{d-1}:{\left(a_j\right)}_{j=1}^d\in {\mathcal{A}}^d,{\displaystyle \sum _{j=1}^d}{a}_j^{*}{a}_j=1\right\}\end{array}$$

and

$$\begin{array}{cc}\hfill w_{\mathcal{A}}\left(R\right)& =sup\left\{\parallel a_2^{*}{a}_1+\dots +a_d^{*}{a}_{d-1}\parallel :{\left(a_j\right)}_{j=1}^d\in {\mathcal{A}}^d,{\displaystyle \sum _{j=1}^d}{a}_j^{*}{a}_j=1\right\}\hfill \\ \hfill & \le 1.\hfill \end{array}$$

Example 6. 

Let $\mathcal{A}$ be a unital C*-algebra and ${\ell }^2\left(\mathcal{A}\right)$ be the standard Hilbert C*-module defined by

$$\begin{array}{c}\hfill {\ell }^2\left(\mathcal{A}\right)\left\{{\left\{a_n\right\}}_{n=1}^{\infty }:a_n\in \mathcal{A},\forall n\in \mathbb{N},{\displaystyle \sum _{n=1}^{\infty }}{a}_n^{*}{a}_n\in \mathcal{A}\right\}\end{array}$$

equipped with inner product

$$\begin{array}{c}\hfill \langle {\left\{a_n\right\}}_{n=1}^{\infty },{\left\{b_n\right\}}_{n=1}^{\infty }\rangle {\displaystyle \sum _{n=1}^{\infty }}{a}_n^{*}{b}_n,\forall {\left\{a_n\right\}}_{n=1}^{\infty },{\left\{b_n\right\}}_{n=1}^{\infty }\in {\ell }^2\left(\mathcal{A}\right)\end{array}$$

and norm

$$\begin{array}{c}\hfill \parallel {\left\{a_n\right\}}_{n=1}^{\infty }\parallel ={∥{\displaystyle \sum _{n=1}^{\infty }}{a}_n^{*}{a}_n∥}^{{\displaystyle \frac{1}{2}}},\forall {\left\{a_n\right\}}_{n=1}^{\infty }\in {\ell }^2\left(\mathcal{A}\right).\end{array}$$

Let L be the left shift morphism defined by

$$\begin{array}{c}\hfill L:{\ell }^2\left(\mathcal{A}\right)\ni {\left\{a_n\right\}}_{n=1}^{\infty }\mapsto L{\left\{a_n\right\}}_{n=1}^{\infty }{\left\{a_{n+1}\right\}}_{n=1}^{\infty }\in {\ell }^2\left(\mathcal{A}\right).\end{array}$$

Then

$$\begin{array}{c}\hfill W_{\mathcal{A}}\left(L\right)=\left\{{\displaystyle \sum _{n=1}^{\infty }}{a}_n^{*}{a}_{n+1}:{\left\{a_n\right\}}_{n=1}^{\infty }\in {\ell }^2\left(\mathcal{A}\right),{\displaystyle \sum _{n=1}^{\infty }}{a}_n^{*}{a}_n=1\right\}\end{array}$$

and

$$\begin{array}{cc}\hfill w_{\mathcal{A}}\left(L\right)& =sup\left\{∥{\displaystyle \sum _{n=1}^{\infty }}{a}_n^{*}{a}_{n+1}∥:{\left\{a_n\right\}}_{n=1}^{\infty }\in {\ell }^2\left(\mathcal{A}\right),{\displaystyle \sum _{n=1}^{\infty }}{a}_n^{*}{a}_n=1\right\}\hfill \\ \hfill & \le 1.\hfill \end{array}$$

Example 7. 

Define

$$\begin{array}{c}\hfill A\left(\begin{array}{cc}a& b\\ c& d\end{array}\right).\end{array}$$

Then

$$\begin{array}{cc}\hfill W_{\mathcal{A}}\left(A\right)& =\{x^{*}ax+x^{*}by+y^{*}cx+y^{*}dy:x,y\in \mathcal{A},x^{*}x+y^{*}y=1\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill w_{\mathcal{A}}\left(A\right)& =sup\{\parallel x^{*}ax+x^{*}by+y^{*}cx+y^{*}dy\parallel :x,y\in \mathcal{A},x^{*}x+y^{*}y=1\}\hfill \\ \hfill & \le \parallel a\parallel +\parallel b\parallel +\parallel c\parallel +\parallel d\parallel .\hfill \end{array}$$

Given $r>0$ and $v\in \mathcal{A}$, the closed disc centered at $v\in \mathcal{A}$ of radius r is defined as

$$\begin{array}{c}\hfill {\mathbb{D}}_{\mathcal{A}}[v,r]\{z\in \mathcal{A}:\parallel z-v\parallel \le r\}.\end{array}$$

Following are basic properties of noncommutative numerical range and radius.

Proposition 1. 

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 $\mathcal{E}$.

(i)

$W_{\mathcal{A}}\left(T\right)\subseteq {\mathbb{D}}_{\mathcal{A}}[0,\parallel T\parallel ]$.

(ii)

$0\le w_{\mathcal{A}}\left(T\right)\le \parallel T\parallel$.

(iii)

$W_{\mathcal{A}}(T+S)\subseteq W_{\mathcal{A}}\left(T\right)+W_{\mathcal{A}}\left(S\right)$.

(iv)

$w_{\mathcal{A}}(S+T)\le w_{\mathcal{A}}\left(S\right)+w_{\mathcal{A}}\left(T\right)$.

(v)

$W_{\mathcal{A}}\left(Ta\right)=W_{\mathcal{A}}\left(T\right)a$.

(vi)

$w_{\mathcal{A}}\left(Ta\right)\le \parallel a\parallel w_{\mathcal{A}}\left(T\right)$.

(vii)

$W_{\mathcal{A}}\left(T^{*}\right)={\left(W_{\mathcal{A}}\left(T\right)\right)}^{*}$.

(viii)

$w_{\mathcal{A}}\left(T^{*}\right)=w_{\mathcal{A}}\left(T\right)$.

(ix)

$W_{\mathcal{A}}(Ta+bI)=W_{\mathcal{A}}\left(T\right)a+b$.

(x)

$W_{\mathcal{A}}(Re\left(T\right))=Re\left(W_{\mathcal{A}}\left(T\right)\right)$ and $W_{\mathcal{A}}(Im\left(T\right))=Im\left(W_{\mathcal{A}}\left(T\right)\right)$.

(xi)

If $T\ge 0$, then $W_{\mathcal{A}}\left(T\right)\subseteq {\mathcal{A}}^{+}$.

(xii)

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 $W_{\mathcal{A}}\left(T\right)\subseteq W_{\mathcal{A}}\left(V\right)$.

(xiii)

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 $W_{\mathcal{A}}\left(P{T}_{|{\mathcal{E}}_1}\right)\subseteq W_{\mathcal{A}}\left(T\right)$.

(xiv)

$W_{\mathcal{A}}\left(U^{*}TU\right)=W_{\mathcal{A}}\left(T\right)$ for every unitary morphism $U:\mathcal{E}\to \mathcal{E}$.

(xv)

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 $a\in W_{\mathcal{A}}\left(T\right)$. In other words,

$$\begin{array}{c}\hfill {\sigma }_1\left(T\right)\{a\in \mathcal{A}:\exists x\in \mathcal{E},Tx=xa,\langle x,x\rangle =1\}\subseteq W_{\mathcal{A}}\left(T\right)\end{array}$$

and

$$\begin{array}{c}\hfill \parallel a\parallel \le w_{\mathcal{A}}\left(T\right),\forall a\in {\sigma }_1\left(T\right).\end{array}$$

(xvi)

$$\begin{array}{c}\hfill w_{\mathcal{A}}(ST+TS)\le {\displaystyle \frac{w_{\mathcal{A}}\left({(S+T)}^2\right)+w_{\mathcal{A}}\left({(S-T)}^2\right)}{2}}.\end{array}$$

In particular, if $ST=TS$, then

$$\begin{array}{c}\hfill w_{\mathcal{A}}\left(ST\right)\le {\displaystyle \frac{w_{\mathcal{A}}\left({(S+T)}^2\right)+w_{\mathcal{A}}\left({(S-T)}^2\right)}{4}}.\end{array}$$

(xvii)

If a sequence ${\left\{T_n\right\}}_{n=1}^{\infty }$ of adjointable morphisms on $\mathcal{E}$ converges to an adjointable morphism T on $\mathcal{E}$ in the morphism norm, then the sequence ${\left\{w_{\mathcal{A}}\left(T_n\right)\right\}}_{n=1}^{\infty }$ converges to $w_{\mathcal{A}}\left(T\right)$ in the norm.

Using generalized polarization identity, it is known that $w_{\mathbb{C}}\left(T\right)\ge \parallel T\parallel /2$ [5]. We are unable to derive noncommutative version of this result. For $\tau ,\omega \in \mathcal{E}$, define the morphism

$$\begin{array}{c}\hfill \tau \otimes \omega :\mathcal{E}\ni x\mapsto (\tau \otimes \omega )x\tau \langle \omega ,x\rangle \in \mathcal{E}.\end{array}$$

Following is modular version of Theorem 3.

Theorem 13. 

Let $\mathcal{E}$ be a Hilbert C*-module over a commutative unital C*-algebra $\mathcal{A}$ and $\tau ,\omega \in \mathcal{E}$. Then

$$\begin{array}{c}\hfill w_{\mathcal{A}}(\tau \otimes \omega )\le {\displaystyle \frac{\parallel \langle \tau ,\omega \rangle \parallel +\parallel \tau \parallel \parallel \omega \parallel }{2}}\le \parallel \tau \parallel \parallel \omega \parallel .\end{array}$$

(7)

In particular,

$$\begin{array}{c}\hfill w_{\mathcal{A}}(\tau \otimes \tau )\le {\parallel \tau \parallel }^2.\end{array}$$

Proof. 

Let $\tau ,\omega \in \mathcal{E}$. Let $x\in \mathcal{E}$ with $\langle x,x\rangle =1$. Then using commutativity of C*-algebra,

$$\begin{array}{cc}\hfill \parallel \langle x,(\tau \otimes \omega )x\rangle \parallel & =\parallel \langle x,\tau \langle \omega ,x\rangle \rangle \parallel =\parallel \langle x,\tau \rangle \langle \omega ,x\rangle \parallel \hfill \\ \hfill & =\parallel \langle \omega ,x\rangle \langle x,\omega \rangle \parallel \le {\displaystyle \frac{\parallel \langle \tau ,\omega \rangle \parallel +\parallel \tau \parallel \parallel \omega \parallel }{2}}.\hfill \end{array}$$

Therefore

$$\begin{array}{c}\hfill w_{\mathcal{A}}(\tau \otimes \omega )\le {\displaystyle \frac{\parallel \langle \tau ,\omega \rangle \parallel +\parallel \tau \parallel \parallel \omega \parallel }{2}}.\end{array}$$

□

Unlike the Hilbert space case, we are unable to derive equality in Inequality (7). Let $\mathcal{A}$ be a unital C*-algebra and $p\left(z\right)a_0+a_{1}z+\dots +a_{n-1}{z}^{n-1}+z^n\in \mathcal{A}\left[z\right]$. Define the modular Frobenius companion matrix

$$\begin{array}{c}\hfill C_p\left(\begin{array}{ccccccc}-a_{n-1}& -a_{n-2}& -a_{n-3}& \dots & -a_2& -a_1& -a_0\\ 1& 0& 0& \dots & 0& 0& 0\\ 0& 1& 0& \dots & 0& 0& 0\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & 0& 0& 0\\ 0& 0& 0& \dots & 1& 0& 0\\ 0& 0& 0& \dots & 0& 1& 0\end{array}\right)\in {\mathbb{M}}_n\left(\mathcal{A}\right).\end{array}$$

Given $n\in \mathbb{N}$, we consider ${\mathcal{A}}^n$ with the standard inner product

$$\begin{array}{c}\hfill \langle {\left(a_j\right)}_{j=1}^n,{\left(b_j\right)}_{j=1}^n\rangle {\displaystyle \sum _{j=1}^n}{a}_j^{*}{b}_j,\forall {\left(a_j\right)}_{j=1}^n,{\left(b_j\right)}_{j=1}^n\in {\mathcal{A}}^n.\end{array}$$

Hence the norm on ${\mathcal{A}}^n$ is

$$\begin{array}{c}\hfill \parallel {\left(a_j\right)}_{j=1}^n\parallel ={∥{\displaystyle \sum _{j=1}^n}{a}_j^{*}{a}_j∥}^{{\displaystyle \frac{1}{2}}},\forall {\left(a_j\right)}_{j=1}^n\in {\mathcal{A}}^n.\end{array}$$

We then have the following result.

Theorem 14. 

Let $\mathcal{A}$ be a commutative unital C*-algebra and $p\left(z\right)a_0+a_{1}z+\dots +a_{n-1}{z}^{n-1}+z^n\in \mathcal{A}\left[z\right]$. If $b\in \mathcal{A}$ satisfies $p\left(b\right)=0$, then $b\in W_{\mathcal{A}}\left(C_p\right)$.

Proof. 

Define

$$\begin{array}{c}\hfill x\left(\begin{array}{c}{b}^{n-1}\\ b^{n-2}\\ b^{n-3}\\ ⋮\\ b^2\\ b\\ 1\end{array}\right){\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}.\end{array}$$

Then $\langle x,x\rangle =1$. Using commutativity of C*-algebra, we have

$$\begin{array}{cc}\hfill & \langle x,C_{p}x\rangle =\hfill \\ \hfill & 〈\left(\begin{array}{c}{b}^{n-1}\\ b^{n-2}\\ b^{n-3}\\ ⋮\\ b^2\\ b\\ 1\end{array}\right){\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}},\left(\begin{array}{ccccccc}-a_{n-1}& -a_{n-2}& -a_{n-3}& \dots & -a_2& -a_1& -a_0\\ 1& 0& 0& \dots & 0& 0& 0\\ 0& 1& 0& \dots & 0& 0& 0\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & 0& 0& 0\\ 0& 0& 0& \dots & 1& 0& 0\\ 0& 0& 0& \dots & 0& 1& 0\end{array}\right)\left(\begin{array}{c}{b}^{n-1}\\ b^{n-2}\\ b^{n-3}\\ ⋮\\ b^2\\ b\\ 1\end{array}\right){\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}〉\hfill \\ \hfill & ={\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}〈\left(\begin{array}{c}{b}^{n-1}\\ b^{n-2}\\ b^{n-3}\\ ⋮\\ b^2\\ b\\ 1\end{array}\right),\left(\begin{array}{ccccccc}-a_{n-1}& -a_{n-2}& -a_{n-3}& \dots & -a_2& -a_1& -a_0\\ 1& 0& 0& \dots & 0& 0& 0\\ 0& 1& 0& \dots & 0& 0& 0\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & 0& 0& 0\\ 0& 0& 0& \dots & 1& 0& 0\\ 0& 0& 0& \dots & 0& 1& 0\end{array}\right)\left(\begin{array}{c}{b}^{n-1}\\ b^{n-2}\\ b^{n-3}\\ ⋮\\ b^2\\ b\\ 1\end{array}\right)〉{\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}\hfill \\ \hfill & ={\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}〈\left(\begin{array}{c}{b}^{n-1}\\ b^{n-2}\\ b^{n-3}\\ ⋮\\ b^2\\ b\\ 1\end{array}\right),\left(\begin{array}{c}-{\displaystyle \sum _{j=0}^{n-1}}{a}_j{b}^j\\ b^{n-1}\\ b^{n-2}\\ ⋮\\ b^3\\ b^2\\ b\end{array}\right)〉{\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}\hfill \\ \hfill & ={\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}〈\left(\begin{array}{c}{b}^{n-1}\\ b^{n-2}\\ b^{n-3}\\ ⋮\\ b^2\\ b\\ 1\end{array}\right),\left(\begin{array}{c}{b}^n\\ b^{n-1}\\ b^{n-2}\\ ⋮\\ b^3\\ b^2\\ b\end{array}\right)〉{\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}\hfill \\ \hfill & ={\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}〈\left(\begin{array}{c}{b}^{n-1}\\ b^{n-2}\\ b^{n-3}\\ ⋮\\ b^2\\ b\\ 1\end{array}\right),\left(\begin{array}{c}{b}^{n-1}\\ b^{n-2}\\ b^{n-3}\\ ⋮\\ b^2\\ b\\ 1\end{array}\right)〉b{\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}\hfill \\ \hfill & ={\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)b{\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}\hfill \\ \hfill & ={\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right){\left({\displaystyle \sum _{j=0}^{n-1}}{\left(b^{*}\right)}^j{b}^j\right)}^{{\displaystyle \frac{-1}{2}}}b\hfill \\ \hfill & =b.\hfill \end{array}$$

Therefore $b=\langle x,C_{p}x\rangle \in W_{\mathcal{A}}\left(C_p\right)$. □

We now derive modular analogue of Theorem 4.

Theorem 15. 

Let $\mathcal{A}$ be a commutative unital C*-algebra and $p\left(z\right)a_0+a_{1}z+\dots +a_{n-1}{z}^{n-1}+z^n\in \mathcal{A}\left[z\right]$. If $b\in \mathcal{A}$ satisfies $p\left(b\right)=0$, then

$$\begin{array}{cc}\hfill \parallel b\parallel & \le w_{\mathcal{A}}\left(R\right)+{\displaystyle \frac{\sqrt{∥{\sum }_{j=0}^{n-1}{a}_j^{*}{a}_j∥}+\parallel a_{n-1}\parallel }{2}}\hfill \\ \hfill & \le w_{\mathcal{A}}\left(R\right)+{\displaystyle \frac{\sqrt{{\sum }_{j=0}^{n-1}{\parallel a_j\parallel }^2}+\parallel a_{n-1}\parallel }{2}},\hfill \end{array}$$

where R is the right shift matrix defined by

$$\begin{array}{c}\hfill R\left(\begin{array}{ccccccc}0& 0& 0& \dots & 0& 0& 0\\ 1& 0& 0& \dots & 0& 0& 0\\ 0& 1& 0& \dots & 0& 0& 0\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & 0& 0& 0\\ 0& 0& 0& \dots & 1& 0& 0\\ 0& 0& 0& \dots & 0& 1& 0\end{array}\right)\in {\mathbb{M}}_n\left(\mathcal{A}\right).\end{array}$$

Proof. 

Define

$$\begin{array}{c}\hfill \mathbf{a}\left(\begin{array}{c}{\left(a_{n-1}\right)}^{*}\\ {\left(a_{n-2}\right)}^{*}\\ {\left(a_{n-3}\right)}^{*}\\ ⋮\\ {\left(a_2\right)}^{*}\\ {\left(a_1\right)}^{*}\\ {\left(a_0\right)}^{*}\end{array}\right)\in {\mathcal{A}}^n,{\mathbf{e}}_{\mathbf{1}}\left(\begin{array}{c}1\\ 0\\ 0\\ ⋮\\ 0\\ 0\\ 0\end{array}\right)\in {\mathcal{A}}^n.\end{array}$$

Then

$$\begin{array}{c}\hfill C_p=R-{\mathbf{e}}_{\mathbf{1}}\otimes \mathbf{a}.\end{array}$$

(8)

Using (8) and Inequality (7), we get

$$\begin{array}{cc}\hfill \parallel b\parallel & \le w_{\mathcal{A}}\left(C_p\right)\hfill \\ \hfill & =w_{\mathcal{A}}(R-{\mathbf{e}}_{\mathbf{1}}\otimes \mathbf{a})\hfill \\ \hfill & \le w_{\mathcal{A}}\left(R\right)+w_{\mathcal{A}}({\mathbf{e}}_{\mathbf{1}}\otimes \mathbf{a})\hfill \\ \hfill & \le w_{\mathcal{A}}\left(R\right)+{\displaystyle \frac{\sqrt{∥{\sum }_{j=0}^{n-1}{a}_j^{*}{a}_j∥}+\parallel a_{n-1}\parallel }{2}}\hfill \\ \hfill & \le w_{\mathcal{A}}\left(R\right)+{\displaystyle \frac{\sqrt{{\sum }_{j=0}^{n-1}{\parallel a_j\parallel }^2}+\parallel a_{n-1}\parallel }{2}}.\hfill \end{array}$$

□

Let $\mathcal{A}$ be a commutative unital C*-algebra. Let $p\left(z\right)a_0+a_{1}z+\dots +a_{n-1}{z}^{n-1}+z^n\in \mathcal{A}\left[z\right]$ with $a_0$ invertible. Define

$$\begin{array}{c}\hfill q\left(z\right){\displaystyle \frac{1}{a_0}}{z}^{n}p\left({\displaystyle \frac{1}{z}}\right)={\displaystyle \frac{1}{a_0}}+{\displaystyle \frac{a_{n-1}}{a_0}}z+\dots +{\displaystyle \frac{a_1}{a_0}}{z}^{n-1}+z^n\in \mathcal{A}\left[z\right].\end{array}$$

We note that an invertible element $b\in \mathcal{A}$ satisfies $p\left(b\right)=0$ if and only if $q(1/b)=0$. Frobenius companion matrix of q is

$$\begin{array}{c}\hfill C_q\left(\begin{array}{ccccccc}-{\displaystyle \frac{a_1}{a_0}}& -{\displaystyle \frac{a_2}{a_0}}& -{\displaystyle \frac{a_3}{a_0}}& \dots & -{\displaystyle \frac{a_{n-1}}{a_0}}& -{\displaystyle \frac{a_{n-1}}{a_0}}& -{\displaystyle \frac{1}{a_0}}\\ 1& 0& 0& \dots & 0& 0& 0\\ 0& 1& 0& \dots & 0& 0& 0\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & 0& 0& 0\\ 0& 0& 0& \dots & 1& 0& 0\\ 0& 0& 0& \dots & 0& 1& 0\end{array}\right)\in {\mathbb{M}}_n\left(\mathcal{A}\right).\end{array}$$

By applying Theorem 4 to q, we get following result.

Theorem 16. 

Let $\mathcal{A}$ be a commutative unital C*-algebra and $p\left(z\right)a_0+a_{1}z+\dots +a_{n-1}{z}^{n-1}+z^n\in \mathcal{A}\left[z\right]$ with $a_0$ invertible. If an invertible element $b\in \mathcal{A}$ satisfies $p\left(b\right)=0$, then

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\parallel b^{-1}\parallel }}\ge {\displaystyle \frac{2}{2w_{\mathcal{A}}\left(R\right)+\sqrt{{∥{\left(a_0^{-1}\right)}^{*}{a}_0^{-1}+{\sum }_{j=1}^{n-1}{\left(a_j{a}_0^{-1}\right)}^{*}{a}_j{a}_0^{-1}∥}^2}+\parallel a_1{a}_0^{-1}\parallel }}.\end{array}$$

Equality term in (4) comes from the numerical radius of right shift matrix obtained by Davidson and Holbrook [21] (also see [22,23]). We are unable to do this for the right shift matrix over C*-algebras. We now derive Theorem 6 for Hilbert C*-modules.

Theorem 17.(Modular Dragomir Numerical Radius Inequality) 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

$$\begin{array}{c}\hfill w_{\mathcal{A}}\left(T\right)\le {\displaystyle \frac{1}{\sqrt{2}}}\sqrt{w_{\mathcal{A}}\left(T^2\right)+{\parallel T\parallel }^2}\le \parallel T\parallel .\end{array}$$

Proof. 

Let $x\in \mathcal{E}$ with $\langle x,x\rangle =1$. Then using Inequality (6) and the self-adjointness of T, we get

$$\begin{array}{cc}\hfill {\parallel \langle x,Tx\rangle \parallel }^2& {=\parallel \langle x,Tx\rangle \langle x,Tx\rangle }^{*}\parallel =\parallel \langle x,Tx\rangle \langle Tx,x\rangle \parallel \hfill \\ \hfill & =\parallel \langle Tx,x\rangle \langle x,Tx\rangle \parallel \le {\displaystyle \frac{\parallel \langle Tx,Tx\rangle \parallel +\parallel Tx\parallel \parallel Tx\parallel }{2}}\hfill \\ \hfill & ={\displaystyle \frac{\parallel \langle x,T^{2}x\rangle \parallel +\parallel Tx\parallel \parallel T^{*}x\parallel }{2}}\le {\displaystyle \frac{\parallel \langle x,T^{2}x\rangle \parallel +\parallel T\parallel \parallel x\parallel \parallel T\parallel \parallel x\parallel }{2}}\hfill \\ \hfill & ={\displaystyle \frac{\parallel \langle x,T^{2}x\rangle {\parallel +\parallel T\parallel }^2}{2}}\le {\displaystyle \frac{{sup}_{y\in \mathcal{E},\langle y,y\rangle =1}\parallel \langle y,T^{2}y\rangle {\parallel +\parallel T\parallel }^2}{2}}\hfill \\ \hfill & ={\displaystyle \frac{w_{\mathcal{A}}\left(T^2\right)+{\parallel T\parallel }^2}{2}}.\hfill \end{array}$$

Therefore

$$\begin{array}{c}\hfill w_{\mathcal{A}}\left(T\right)=\underset{x\in \mathcal{E},\langle x,x\rangle =1}{sup}\parallel \langle x,Tx\rangle \parallel \le {\displaystyle \frac{1}{\sqrt{2}}}\sqrt{w_{\mathcal{A}}\left(T^2\right)+{\parallel T\parallel }^2}.\end{array}$$

□

Note that we derived Theorem 17 only for self-adjoint morphisms. We are unable to derive Theorem 17 for arbitrary adjointable morphisms. We also note that in the case of self-adjoint operator T on Hilbert spaces, we have $w_{\mathbb{C}}\left(T\right)=\parallel T\parallel$ [5] which we can’t say for self-adjoint morphisms on Hilbert C*-modules.

3. Noncommutative Toeplitz-Hausdorff Problem

Motivated from several breakthrough results in numerical range and radius, we formulate following problems. What is noncommutative

(1)

Toeplitz-Hausdorff Theorem (fundamental theorem of the numerical range) [5,6,24,25,26,27,28,29,30,31,32,33]?

(2)

Wintner spectral inclusion theorem for numerical range [5,34,35]?

(3)

Halmos-Bernau-Smithies-Berger power inequality [36,37]?

(4)

Alpin-Chien-Yeh inequality[38]?

(5)

Kittaneh inequalities [39,40]?

(6)

Yamazaki inequality [41]?

(7)

Kittaneh-Moslehian-Yamazaki inequality [42]?

(8)

Abu-Omar-Kittaneh inequality [43]?

(9)

Bhunia-Bag-Paul inequality [44]?

(10)

elliptical range theorem [45,46]?

(11)

Berger power dilation for numerical contraction [5,47,48,49]?

(12)

Ando theorem for numerical contraction [50]?

(13)

Furuta-Nakamoto theorem for numerical contractions [51]?

(14)

El-Haddad-Kittaneh numerical radius inequality [52]?

(15)

description of numerical range of 3 × 3 matrices [53]?

(16)

description of numerical range of 4 × 4 matrices [54]?

(17)

Holbrook numerical radius inequalities for the product of commuting matrices [55]?

(18)

Okubo-Ando numerical radius inequalities for the product of commuting matrices [56,57,58]?

(19)

Bouldin numerical radius inequalities for the product of commuting matrices [59,60]?

(20)

Poncelet property for numerical ranges [61,62,63]?

(21)

Anderson theorem [8,64,65]?

(22)

Kippenhahn boundary generating curve theorem [66,67]?

(23)

Kippenhahn corner-point theorem [66,67]?

(24)

Hildebrandt normal eigenvalue theorem [68,69]?

(25)

Hildebrandt intersection theorem [70,71]?

(26)

Marcus-Shure description for the numerical range of zero-one matrices [72] and Davidson-Holbrook estimate for the numerical radius of zero-one matrices [21]?

(27)

Johnson numerical range inclusion theorem [73]?

(28)

Choi-Li constrained numerical range theorem [74]?

(29)

Wang-Wu-Gau result for Crawford number [75]?

(30)

Abu-Omar-Kittaneh inequality (involving generalized Aluthge transform) [76]?

(31)

Gau-Wu numerical radius equality characterizations [77]?

(32)

Gau-Wu theorem for compact operators [78]?

(33)

Perron-Frobenius type results on the numerical range [79]?

(34)

Helton-Spitkovsky characterization of shapes of numerical ranges [80]?

(35)

Radjabalipour-Radjavi result on numerical ranges [81]?

(36)

Pollack theorem on numerical ranges [82]?

(37)

Anderson result on numerical ranges [83]?

(38)

Abu-Omar-Kittaneh numerical radius inequalities for the product of matrices [84]?

(39)

Kittaneh-Moradi inequality [85]?

(40)

Lancaster characterization for the closedness of numerical range [86]?

(41)

Gau-Wu characterization for numerical ranges of completely non-unitary contractions [87]?

(42)

Mees-Atherton domains containing numerical ranges [88]?

(43)

Chien-Tam characterizations of circularity of numerical ranges [89]?

(44)

Shiu theorem on the growth of numerical range of powers of operator [90]?

4. Noncommutative Crouzeix Inequality, von Neumann Inequality and Ando Inequality Problems

Breakthrough Crouzeix theorem says following.

Theorem 18. 

[91,92,93,94,95] (Crouzeix Theorem) For every $d\in \mathbb{N}$ and for every matrix $M\in {\mathbb{M}}_d\left(\mathbb{C}\right)$, we have

$$\begin{array}{c}\hfill \left(\mathbf{Crouzeix}-\mathbf{Palencia}\mathbf{Inequality}\right)\parallel p\left(M\right)\parallel \le (1+\sqrt{2})sup\left\{\left|p\left(z\right)\right|:z\in W_{\mathbb{C}}\left(M\right)\right\},\forall p\in \mathbb{C}\left[z\right].\end{array}$$

We formulate following problem.

Problem 19.(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 $M\in {\mathbb{M}}_d\left(\mathcal{A}\right)$, we have

$$\begin{array}{c}\hfill \parallel p\left(M\right)\parallel \le R_{\mathcal{A}}sup\left\{\parallel p\left(z\right)\parallel :z\in W_{\mathcal{A}}\left(M\right)\right\},\forall p\in \mathcal{A}\left[z\right].\end{array}$$

An inequality which is very close to Crouzeix inequality is the von Neumann inequality.

Theorem 20. 

[49,96,97,98,99] (von Neumann Theorem) For every $d\in \mathbb{N}$ and for every matrix $M\in {\mathbb{M}}_d\left(\mathbb{C}\right)$ with $\parallel M\parallel \le 1$, we have

$$\begin{array}{c}\hfill \left(\mathbf{von}\mathbf{Neumann}\mathbf{Inequality}\right)\parallel p\left(M\right)\parallel \le sup\left\{\left|p\right(z\left)\right|:z\in \mathbb{C},\left|z\right|\le 1\right\},\forall p\in \mathbb{C}\left[z\right].\end{array}$$

Based on Theorem 20 we formulate following problems.

Problem 21.(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 $A\in {\mathbb{M}}_d\left(\mathcal{A}\right)$ with $\parallel A\parallel \le 1$, we have

$$\begin{array}{c}\hfill \parallel p\left(A\right)\parallel \le R_{\mathcal{A}}sup\left\{\parallel p\left(z\right)\parallel :z\in \mathcal{A},\parallel z\parallel \le 1\right\},\forall p\in \mathcal{A}\left[z\right].\end{array}$$

Theorem 20 has been extended by Ando for two variables.

Theorem 22. 

[49,50] (Ando Theorem) For every $d\in \mathbb{N}$ and for all matrices $M,N\in {\mathbb{M}}_d\left(\mathbb{C}\right)$ with $\parallel M\parallel \le 1$, $\parallel N\parallel \le 1$ and $MN=NM$, we have

$$\begin{array}{c}\hfill \left(\mathbf{Ando}\mathbf{Inequality}\right)\parallel p(M,N)\parallel \le sup\left\{\left|p\right(z,w\left)\right|:z,w\in \mathbb{C},\left|z\right|\le 1,\left|w\right|\le 1\right\},\forall p\in \mathbb{C}[z,w].\end{array}$$

Based on Theorem 22, we formulate following problem.

Problem 23.(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 $A,B\in {\mathbb{M}}_d\left(\mathcal{A}\right)$ with $\parallel A\parallel \le 1$, $\parallel B\parallel \le 1$ and $AB=BA$, we have

$$\begin{array}{c}\hfill \parallel p(A,B)\parallel \le sup\left\{\parallel p(z,w)\parallel :z,w\in \mathcal{A},\parallel z\parallel \le 1,\parallel w\parallel \le 1\right\},\forall p\in \mathcal{A}[z,w].\end{array}$$

Note that Ando theorem cannot be extended to more than two commuting matrices [100,101]. We observe that the Halmos dilation [102], Egervary dilation [103,104] and Sz.-Nagy dilation (Schaffer construction) [105] carry over to adjointable morphisms on Hilbert C*-modules. This observation will give following result.

Theorem 24.(Noncommutative Sz.-Nagy Dilation) 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 $\parallel T\parallel \le 1$. 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

$$\begin{array}{c}\hfill T^{n}x=P_{\mathcal{E}}{U}^{n}x,\forall n\in \mathbb{N},\forall x\in \mathcal{E}.\end{array}$$

Corollary 2. 

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 $\parallel T\parallel \le 1$. Let ${\mathcal{E}}_0$, $P_{\mathcal{E}}$ and U be as in Theorem 24. Then

$$\begin{array}{c}\hfill \parallel p\left(T\right)\parallel \le \parallel p\left(U\right)\parallel ,\forall p\left(z\right)=a_0+z{a}_1+\dots +z^n{a}_n\in \mathcal{A}\left[z\right].\end{array}$$

Corollary 3. 

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 $\parallel T\parallel \le 1$. Let $x\in \mathcal{E}$ be such that $Tx=x$. Then $T^{*}x=x$.

Proof. 

Let $x\in \mathcal{E}$ be such that $Tx=x$. Let ${\mathcal{E}}_0$, $P_{\mathcal{E}}$ and U be as in Theorem 24. We then have

$$\begin{array}{c}\hfill x=Tx=P_{\mathcal{E}}Ux,\langle Ux,Ux\rangle =\langle x,x\rangle ,P_{\mathcal{E}}x=x.\end{array}$$

We see that

$$\begin{array}{cc}\hfill \langle Ux-x,Ux-x\rangle & =2\langle x,x\rangle -\langle Ux,x\rangle -\langle x,Ux\rangle \hfill \\ \hfill & =2\langle x,x\rangle -\langle Ux,P_{\mathcal{E}}x\rangle -\langle P_{\mathcal{E}}x,Ux\rangle \hfill \\ \hfill & =2\langle x,x\rangle -\langle P_{\mathcal{E}}Ux,x\rangle -\langle x,P_{\mathcal{E}}Ux\rangle \hfill \\ \hfill & =2\langle x,x\rangle -\langle Tx,x\rangle -\langle x,Tx\rangle \hfill \\ \hfill & =2\langle x,x\rangle -\langle x,x\rangle -\langle x,x\rangle =0.\hfill \end{array}$$

Therefore

$$\begin{array}{c}\hfill Ux=x\Rightarrow x=U^{*}x.\end{array}$$

Hence

$$\begin{array}{c}\hfill x=U^{*}x=P_{\mathcal{E}}{U}^{*}x=T^{*}x.\end{array}$$

□

We also observe that the proof of Ando dilation [50] 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).