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Noncommutative Buzano-Dragomir Inequality and Applications

Published Oct 07, 2026Fresh — added to the Hub within the last 30 daysv110 min read

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Abstract

Dragomir [Bull. Aust. Math. Soc., 2016] showed that the Buzano inequality holds for orthogonal projections on Hilbert spaces. Dragomir [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.

Introduction

More than half-century ago, Buzano derived the following generalization of the Cauchy-Schwarz inequality [18].

Theorem 1.1. [18, 91] [36] (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

|\langle\tau,h\rangle\langle h,\omega\rangle| \leq\frac{\|h\|^2\left(|\langle\tau,\omega\rangle|+\|\tau\|\|\omega\|\right)}{2} \leq\|h\|^2\|\tau\|\|\omega\|,\quad\forall h,\tau,\omega\in\mathcal{H}. \tag*{(1)}

In other words,

\left|\left\langle\tau,\frac{h}{\lVert h\rVert}\right\rangle\left\langle\frac{h}{\lVert h\rVert},\omega\right\rangle\right|\le\frac{|\langle\tau,\omega\rangle|+\lVert\tau\rVert\lVert\omega\rVert}{2}\le\lVert\tau\rVert\lVert\omega\rVert,\quad\forall\tau,\omega\in\mathcal{H},\ \forall h\in\mathcal{H}\setminus\{0\}.

Further, we have following.

(i) 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

h_{\alpha,\beta}:=\alpha\left(\frac{\tau}{\lVert\tau\rVert}+\beta\frac{\omega}{\lVert\omega\rVert}\right).

Then

\left|\langle\tau,h_{\alpha,\beta}\rangle\langle h_{\alpha,\beta},\omega\rangle\right|=\frac{\lVert h_{\alpha,\beta}\rVert^2\left(|\langle\tau,\omega\rangle|+\lVert\tau\rVert\lVert\omega\rVert\right)}{2}.

(ii) Let $\tau,\omega\in\mathcal{H}\setminus\{0\}$ be such that $\langle\tau,\omega\rangle\ne0$ . For $\alpha\in\mathbb{C}$ , define

h_\alpha:=\alpha\left(\frac{\tau}{\lVert\tau\rVert}+\frac{\langle\tau,\omega\rangle}{|\langle\tau,\omega\rangle|}\frac{\omega}{\lVert\omega\rVert}\right).

Then

\left|\langle\tau,h_\alpha\rangle\langle h_\alpha,\omega\rangle\right|=\frac{\lVert h_\alpha\rVert^2\left(|\langle\tau,\omega\rangle|+\lVert\tau\rVert\lVert\omega\rVert\right)}{2}.

Ten years ago, Dragomir made the following far reaching generalization of Theorem 1.1 [29].

Theorem 1.2. [29] (Buzano-Dragomir Inequality) Let $\mathcal{H}$ be a complex Hilbert space and $P:\mathcal{H}\to\mathcal{H}$ be an orthogonal projection. Then

|\langle P\tau,\omega\rangle|\le\frac{|\langle\tau,\omega\rangle|+\lVert\tau\rVert\lVert\omega\rVert}{2},\quad\forall\tau,\omega\in\mathcal{H}. \tag*{(2)}

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

\mathcal{H}\ni\tau\mapsto\left\langle\tau,\frac{h}{\lVert h\rVert}\right\rangle\frac{h}{\lVert h\rVert}\in\mathcal{H}

is a rank one orthogonal projection. Thus Inequality (2) recovers Inequality (1). Inequality (1) has various applications, we describe few. Recall that [93] [44, 46, 50, 56] the numerical range of a bounded linear operator $T:\mathcal{H}\to\mathcal{H}$ is defined as

W_{\mathbb{C}}(T):=\{\langle Th,h\rangle:h\in\mathcal{H},\ \lVert h\rVert=1\}

and the numerical radius of $T$ is defined as

w_{\mathbb{C}}(T):=\sup_{h\in\mathcal{H},\lVert h\rVert=1}|\langle Th,h\rangle|.

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

\tau\otimes\omega:\mathcal{H}\ni h\mapsto(\tau\otimes\omega)h:=\langle h,\omega\rangle\tau\in\mathcal{H}.

In 1993, Fujii and Kubo derived following result, using Inequality (1) [36].

Theorem 1.3. [36] Let $\mathcal{H}$ be a complex Hilbert space and $\tau,\omega\in\mathcal{H}$ . Then

w_{\mathbb{C}}(\tau\otimes\omega)=\frac{|\langle\tau,\omega\rangle|+\lVert\tau\rVert\lVert\omega\rVert}{2}.

Let $p(z):=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in\mathbb{C}[z]$ . A direct observation reveals that the zeros of $p$ are the eigenvalues of the Frobenius companion matrix

C_p:=\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& \ddots& \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\mathrm{M}_n(\mathbb{C})

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

R:=\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& \ddots& \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\mathrm{M}_n(\mathbb{C}).

Define

\mathbf{a}:=\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 \mathbf{e}_1:=\begin{pmatrix} 1\\ 0\\ 0\\ \vdots\\ 0\\ 0\\ 0 \end{pmatrix}.

Then

C_p=R-\mathbf{e}_1\otimes\mathbf{a}. \tag*{(3)}

Fujii and Kubo derived following bound for the zeros of $p$ using Inequality 1 and the representation (3) [36].

Theorem 1.4. [36] (Fujii-Kubo polynomial root upper bound) Let $p(z):=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in\mathbb{C}[z]$ . If $\lambda$ is a zero of $p$ , then

|\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}. \tag*{(4)}

Let $p(z):=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in\mathbb{C}[z]$ with $a_0\ne0$ . Define

q(z):=\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].

We note that $\lambda\in\mathbb{C} \setminus\{0\}$ satisfies $p(\lambda)=0$ if and only if $q(1/\lambda)=0$ [56]. Frobenius companion matrix of $q$ is

C_q := \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}{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 M_n(\mathbb{C})

By applying Theorem 1.4 to $q$ , we get following result.

Theorem 1.5. [36] (Fujii-Kubo polynomial root lower bound) Let $p(z):=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in\mathbb{C}[z]$ with $a_0\ne0$ . If $\lambda$ is a zero of $p$ , then

|\lambda| \ge\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|}

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

w_{\mathbb{C}}(T)\le\|T\|. \tag*{(5)}

In 2007, Dragomir strengthened Inequality (5) using Buzano inequality [30].

Theorem 1.6. [30] (Dragomir Numerical Radius Inequality) Let $T:\mathcal{H}\to\mathcal{H}$ be a bounded linear operator. Then

w_{\mathbb{C}}(T)\le\frac{1}{\sqrt{2}}\sqrt{w_{\mathbb{C}}(T^2)+\|T\|^2}\le\|T\|.

Our fundamental motivation comes from the following question: What is the noncommutative analogue of Theorem 1.2? This is then naturally connected with the notion of Hilbert C*-modules which are first introduced by Kaplansky [58] for modules over commutative C*-algebras and later developed for modules over arbitrary C*-algebras by Paschke [81] and Rieffel [88].

Definition 1.7. [81, 88] [58] Let $\mathcal{A}$ be a unital C*-algebra. A right module $\mathcal{E}$ over $\mathcal{A}$ is said to be a 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.

(i) $\langle x,x\rangle\ge0$ , $\forall x\in\mathcal{E}$ .

(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 1.8. [68] (Kaplansky-Paschke-Rieffel Inequality) Let $\mathcal{E}$ be a semi-inner product C*-module. Then

\langle y,x\rangle\langle x,y\rangle\le\|\langle x,x\rangle\|\langle y,y\rangle,\quad\forall x,y\in\mathcal{E}.

In particular,

\|\langle y,x\rangle\|^2\le\|\langle x,x\rangle\|\|\langle y,y\rangle\|,\quad\forall x,y\in\mathcal{E}.

Definition 1.9. [68] A semi-inner product $C^*$ -module $\mathcal{E}$ is said to be an inner product $C^*$ -module if $x \in\mathcal{E}$ satisfies $\langle x,x\rangle= 0$ , then $x = 0$ .

Definition 1.10. [68] A inner product $C^*$ -module $\mathcal{E}$ is said to be Hilbert $C^*$ -module if $\mathcal{E}$ is complete w.r.t. the norm $\|x\| := \sqrt{\|\langle x,x\rangle\|}$ , $\forall x \in\mathcal{E}$ .

In 2012, Khosravi, Drnovsek and Moslehian derived noncommutative version of Inequality (1) [61].

Theorem 1.11. [61] (Buzano-Khosravi-Drnovsek-Moslehian Inequality) Let $\mathcal{E}$ be a Hilbert $C^*$ -module. Then

\|\langle\omega,x\rangle\langle x,\tau\rangle\| \le\frac{\|\langle\tau,\omega\rangle\|+\|\tau\|\|\omega\|}{2} \le\|\tau\|\|\omega\|,\qquad\forall\tau,\omega\in\mathcal{E},\ \forall x \in\mathcal{E}\text{ with }\langle x,x\rangle=1. \tag*{(6)}

In this article, we show that Theorem 1.2 extends to orthogonal projections on Hilbert $C^*$ -modules (which will also generalize Theorem 1.11). We also derive modular analogue of Theorems 1.4 and 1.6.

Noncommutative Buzano-Dragomir Inequality

We start by deriving noncommutative analogue of Theorem 1.2.

Theorem 2.1. (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

\|\langle\tau,P\omega\rangle\| \le\frac{\|\langle\tau,\omega\rangle\|+\|\tau\|\|\omega\|}{2},\qquad\forall\tau,\omega\in\mathcal{E}.

Proof. We find

\|2\langle\tau,P\omega\rangle-\langle\tau,\omega\rangle\|=\|\langle\tau,2P\omega-\omega\rangle\|\le\|\tau\|\|2P\omega-\omega\|.

We now find

\begin{aligned} \|2P\omega-\omega\|^2 &= \|\langle2P\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{aligned}

Therefore

2\|\langle\tau,P\omega\rangle\|-\|\langle\tau,\omega\rangle\|\le\|2\langle\tau,P\omega\rangle-\langle\tau,\omega\rangle\|\le\|\tau\|\|\omega\|.

Corollary 2.2. Theorem 1.11 follows from Theorem 2.1.

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

P:\mathcal{E}\ni y\mapsto Py:=x\langle x,y\rangle\in\mathcal{E}.

Then $P$ is an orthogonal projection. Theorem 2.1 gives

\|\langle\tau,x\rangle\langle x,\omega\rangle\|=\|\langle\tau,x\langle x,\omega\rangle\rangle\|=\|\langle\tau,P\omega\rangle\| \le\frac{\|\langle\tau,\omega\rangle\|+\|\tau\|\|\omega\|}{2},\qquad\forall\tau,\omega\in\mathcal{E}.

In 2016, Dragomir further improved Theorem 1.2 for bounded linear operators [28].

Theorem 2.3. [28] (Buzano-Dragomir Inequality) Let $\mathcal{H}$ be a complex Hilbert space and $T:\mathcal{H}\to\mathcal{H}$ be a bounded linear operator. Then

|\langle T\tau,T\omega\rangle|\leq\frac{\lVert T\rVert^{2}(|\langle\tau,\omega\rangle|+\lVert\tau\rVert\lVert\omega\rVert)}{2},\qquad\forall\tau,\omega\in\mathcal{H}.

We now derive following noncommutative version of Theorem 2.3.

Theorem 2.4. (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

\lVert\langle T\tau,T\omega\rangle\rVert\leq\lVert T\rVert^{2}\lVert\langle\tau,\omega\rangle\rVert+\left\lVert\langle\tau,\tau\rangle\lVert T\rVert^{2}-\langle T\tau,T\tau\rangle\right\rVert^{\frac{1}{2}}\left\lVert\langle\omega,\omega\rangle\lVert T\rVert^{2}-\langle T\omega,T\omega\rangle\right\rVert^{\frac{1}{2}},\qquad\forall\tau,\omega\in\mathcal{E}.

Proof. Our proof is motivated by the Hilbert space argument due to Dragomir [28]. Let $T:\mathcal{E}\to\mathcal{E}$ be an adjointable morphism and $\tau,\omega\in\mathcal{E}$ . Define

[\cdot,\cdot]:\mathcal{E}\times\mathcal{E}\ni(x,y)\mapsto[x,y]:=\langle x,y\rangle\lVert T\rVert^{2}-\langle Tx,Ty\rangle\in\mathcal{A}.

Then $[\cdot,\cdot]$ is a semi-inner product. By applying Theorem 1.8 to this semi-inner product, we get

\lVert[x,y]\rVert^{2}\leq\lVert[x,x]\rVert\lVert[y,y]\rVert,\qquad\forall x,y\in\mathcal{E}.

In particular,

\lVert[\tau,\omega]\rVert^{2}\leq\lVert[\tau,\tau]\rVert\lVert[\omega,\omega]\rVert.

Previous inequality gives

\left\lVert\langle\tau,\omega\rangle\lVert T\rVert^{2}-\langle T\tau,T\omega\rangle\right\rVert^{2}\leq\left\lVert\langle\tau,\tau\rangle\lVert T\rVert^{2}-\langle T\tau,T\tau\rangle\right\rVert\left\lVert\langle\omega,\omega\rangle\lVert T\rVert^{2}-\langle T\omega,T\omega\rangle\right\rVert.

Therefore

\begin{aligned} \lVert\langle T\tau,T\omega\rangle\rVert-\lVert T\rVert^{2}\lVert\langle\tau,\omega\rangle\rVert\leq\left\lVert\langle\tau,\omega\rangle\lVert T\rVert^{2}-\langle T\tau,T\omega\rangle\right\rVert\\ &\leq\left\lVert\langle\tau,\tau\rangle\lVert T\rVert^{2}-\langle T\tau,T\tau\rangle\right\rVert^{\frac{1}{2}}\left\lVert\langle\omega,\omega\rangle\lVert T\rVert^{2}-\langle T\omega,T\omega\rangle\right\rVert^{\frac{1}{2}}. \end{aligned}

We are unable to simplify Theorem 2.4. We therefore ask following problem.

Problem 2.5. 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

\lVert\langle T\tau,T\omega\rangle\rVert\leq\frac{\lVert T\rVert^{2}(\lVert\langle\tau,\omega\rangle\rVert+\lVert\tau\rVert\lVert\omega\rVert)}{2},\qquad\forall\tau,\omega\in\mathcal{E}?

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

W_{\mathcal{A}}(T):=\{\langle x,Tx\rangle:x\in\mathcal{E},\langle x,x\rangle=1\}\subseteq\mathcal{A}

and the noncommutative numerical radius $T$ is defined as

w_{\mathcal{A}}(T):=\sup_{x\in\mathcal{E},\langle x,x\rangle=1}\lVert\langle x,Tx\rangle\rVert\geq0.

Note that our definition of numerical range and radius on Hilbert C*-modules differ from the existing same notion on Hilbert C*-modules [76, 86, 105] We give various examples. In all following examples, $\mathcal{A}$ is a unital C*-algebra.

Example 2.6. Let $b \in\mathcal{A}$ . Then

W_{\mathcal{A}}(b)=\{a^{*}ba:a\in\mathcal{A},a^{*}a=1\}

and

w_{\mathcal{A}}(b)=\sup\{\lVert a^{*}ba\rVert:a\in\mathcal{A},a^{*}a=1\}

=\lVert b\rVert.

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

Example 2.7. Let $d\in\mathbb{N}$ and $I_d$ be the identity matrix of size $d$ by $d$ . Then we have

W_{\mathcal{A}}(I_d)=\{\langle x,x\rangle:x\in\mathcal{A}^{d},\langle x,x\rangle=1\}=\{1\}

and

w_{\mathcal{A}}(I_d)=1.

Example 2.8. Define

M:=\begin{pmatrix}0&1\\0&0\end{pmatrix}.

Then

W_{\mathcal{A}}(M)=\{\langle x,Mx\rangle:x\in\mathcal{A}^{2},\langle x,x\rangle=1\}

=\{a^{*}b:a,b\in\mathcal{A},a^{*}a+b^{*}b=1\}

and

w_{\mathcal{A}}(M)=\sup\{\lVert a^{*}b\rVert:a,b\in\mathcal{A},a^{*}a+b^{*}b=1\}.

We note that

a^{*}bb^{*}a\leq\frac{a^{*}a+b^{*}b}{2},\qquad\forall a,b\in\mathcal{A}.

Therefore

w_{\mathcal{A}}(M)=\sup\{\lVert a^{*}b\rVert:a,b\in\mathcal{A},a^{*}a+b^{*}b=1\}

\leq\frac{1}{\sqrt{2}}\sup\{\sqrt{\lVert a^{*}a+b^{*}b\rVert}:a,b\in\mathcal{A},a^{*}a+b^{*}b=1\}

=\frac{1}{\sqrt{2}}.

Example 2.9. Let $d\in\mathbb{N}$ and $b_1,\ldots,b_d\in\mathcal{A}$ . Define $\operatorname{diag}((b_j)_{j=1}^{d})$ as the diagonal matrix with diagonal entries $b_1,\ldots,b_d$ . Then

W_{\mathcal{A}}\left(\operatorname{diag}\left((b_j)_{j=1}^{d}\right)\right) = \left\{ \sum_{j=1}^{d}a_j^{*}b_ja_j:(a_j)_{j=1}^{d}\in\mathcal{A}^{d},\sum_{j=1}^{d}a_j^{*}a_j=1 \right\}

and

\begin{aligned} w_{\mathcal{A}}(\operatorname{diag}((b_j)_{j=1}^{d}))=\sup\left\{\left\|\sum_{j=1}^{d}a_j^{*}b_ja_j\right\|:(a_j)_{j=1}^{d}\in\mathcal{A}^{d},\ \sum_{j=1}^{d}a_j^{*}a_j=1\right\} \\ &\leq\sum_{j=1}^{d}\lVert b_j\rVert. \end{aligned}

The set $W_{\mathcal{A}}(\operatorname{diag}((b_j)_{j=1}^{d}))$ is the $C^{*}$ -convex hull of $\{b_j\}_{j=1}^{d}$ [71].

Example 2.10. Let $R\in\mathbb{M}_{d}(\mathcal{A})$ be the right shift matrix defined by

R:=\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& \ddots& \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}.

Thus the action of $R$ is

R((a_j)_{j=1}^{d})=(0,a_1,a_2,\ldots,a_{d-1}),\quad\forall(a_j)_{j=1}^{d}\in\mathcal{A}^{d}.

We then have

\left\langle(a_j)_{j=1}^{d},R(a_j)_{j=1}^{d}\right\rangle=a_2^{*}a_1+\cdots+a_d^{*}a_{d-1},\quad\forall(a_j)_{j=1}^{d}\in\mathcal{A}^{d}.

Hence

\begin{aligned} \left\|\left\langle(a_j)_{j=1}^{d},R(a_j)_{j=1}^{d}\right\rangle\right\|=\left\|a_2^{*}a_1+\cdots+a_d^{*}a_{d-1}\right\| \\ &\leq\left\|\sum_{j=2}^{d}a_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}^{d}a_j^{*}a_j\right\|^{\frac{1}{2}}\left\|\sum_{k=1}^{d}a_k^{*}a_k\right\|^{\frac{1}{2}} \\ &=\left\|\sum_{j=1}^{d}a_j^{*}a_j\right\|,\quad\forall(a_j)_{j=1}^{d}\in\mathcal{A}^{d}. \end{aligned}

Therefore

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\}

and

w_{\mathcal{A}}(R)=\sup\left\{\left\|a_2^{*}a_1+\cdots+a_d^{*}a_{d-1}\right\|:(a_j)_{j=1}^{d}\in\mathcal{A}^{d},\ \sum_{j=1}^{d}a_j^{*}a_j=1\right\}

\leq1.

Example 2.11. Let $A$ be a unital $C^*$ -algebra and $\ell^2(A)$ be the standard Hilbert $C^*$ -module defined by

\ell^2(A):=\left\{\{a_n\}_{n=1}^{\infty}:a_n\in A,\ \forall n\in\mathbb{N},\ \sum_{n=1}^{\infty}a_n^*a_n\in A\right\}.

equipped with inner product

\left\langle\{a_n\}_{n=1}^{\infty},\{b_n\}_{n=1}^{\infty}\right\rangle:=\sum_{n=1}^{\infty}a_n^*b_n,\quad\forall\{a_n\}_{n=1}^{\infty},\{b_n\}_{n=1}^{\infty}\in\ell^2(A).

and norm

\left\|\{a_n\}_{n=1}^{\infty}\right\|=\left\|\sum_{n=1}^{\infty}a_n^*a_n\right\|^{\frac{1}{2}},\quad\forall\{a_n\}_{n=1}^{\infty}\in\ell^2(A).

Let $L$ be the left shift morphism defined by

L:\ell^2(A)\ni\{a_n\}_{n=1}^{\infty}\mapsto L\{a_n\}_{n=1}^{\infty}:=\{a_{n+1}\}_{n=1}^{\infty}\in\ell^2(A).

Then

W_A(L)=\left\{\sum_{n=1}^{\infty}a_n^*a_{n+1}:\{a_n\}_{n=1}^{\infty}\in\ell^2(A),\ \sum_{n=1}^{\infty}a_n^*a_n=1\right\}.

and

w_A(L)=\sup\left\{\left\|\sum_{n=1}^{\infty}a_n^*a_{n+1}\right\|:\{a_n\}_{n=1}^{\infty}\in\ell^2(A),\ \sum_{n=1}^{\infty}a_n^*a_n=1\right\}\leq1.

Example 2.12. Define

A:=\begin{pmatrix}a&b\\c&d\end{pmatrix}.

Then

W_A(A)=\{x^*ax+x^*by+y^*cx+y^*dy:x,y\in A,\ x^*x+y^*y=1\}.

and

w_A(A)=\sup\{\|x^*ax+x^*by+y^*cx+y^*dy\|:x,y\in A,\ x^*x+y^*y=1\} \leq\|a\|+\|b\|+\|c\|+\|d\|.

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

D_A[v,r]:=\{z\in A:\|z-v\|\leq r\}.

Following are basic properties of noncommutative numerical range and radius.

Proposition 2.13. Let $\mathcal{E}$ be a Hilbert $C^*$ -module over a unital $C^*$ -algebra $A$ . Let $A^+$ be the set of all positive elements in $A$ . Let $T,S:\mathcal{E}\to\mathcal{E}$ be adjointable morphisms and $a,b\in A$ . Let $I$ be the identity morphism on $\mathcal{E}$ .

(i) $W_A(T)\subseteq D_A[0,\|T\|]$ .

(ii) $0\leq w_A(T)\leq\|T\|$ .

(iii) $W_A(T+S)\subseteq W_A(T)+W_A(S)$ .

(iv) $w_A(S+T)\leq w_A(S)+w_A(T)$ .

(v) $W_A(Ta)=W_A(T)a$ .
(vi) $w_A(Ta)\le\lVert a\rVert w_A(T)$ .
(vii) $W_A(T^*)=(W_A(T))^*$ .
(viii) $w_A(T^*)=w_A(T)$ .
(ix) $W_A(Ta+bI)=W_A(T)a+b$ .
(x) $W_A(\operatorname{Re}(T))=\operatorname{Re}(W_A(T))$ and $W_A(\operatorname{Im}(T))=\operatorname{Im}(W_A(T))$ .
(xi) If $T\ge0$ , then $W_A(T)\subseteq A^+$ .
(xii) Let $\mathcal{E}_0$ be a Hilbert $C^*$ -module over a unital $C^*$ -algebra $A$ 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 $W_A(T)\subseteq W_A(V)$ .
(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 $W_A(P T|_{\mathcal{E}_1})\subseteq W_A(T)$ .
(xiv) $W_A(U^*TU)=W_A(T)$ for every unitary morphism $U:\mathcal{E}\to\mathcal{E}$ .
(xv) Let $a\in A$ be such that there exists a $x\in\mathcal{E}$ with $Tx=xa$ and $\langle x,x\rangle=1$ . Then $a\in W_A(T)$ . In other words,

\sigma_1(T):=\{a\in A:\exists x\in\mathcal{E},Tx=xa,\langle x,x\rangle=1\}\subseteq W_A(T)

and

\lVert a\rVert\le w_A(T),\qquad\forall a\in\sigma_1(T).

(xvi)

w_A(ST+TS)\le\frac{w_A((S+T)^2)+w_A((S-T)^2)}{2}.

In particular, if $ST=TS$ , then

w_A(ST)\le\frac{w_A((S+T)^2)+w_A((S-T)^2)}{4}.

(xvii) 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 $\{w_A(T_n)\}_{n=1}^{\infty}$ converges to $w_A(T)$ in the norm.

Using generalized polarization identity, it is known that $w_C(T)\ge\lVert T\rVert/2$ [46]. We are unable to derive noncommutative version of this result. For $\tau,\omega\in\mathcal{E}$ , define the morphism

\tau\otimes\omega:\mathcal{E}\ni x\mapsto(\tau\otimes\omega)x:=\tau\langle\omega,x\rangle\in\mathcal{E}.

Following is modular version of Theorem 1.3.

Theorem 2.14. Let $\mathcal{E}$ be a Hilbert $C^*$ -module over a commutative unital $C^*$ -algebra $A$ and $\tau,\omega\in\mathcal{E}$ . Then

w_A(\tau\otimes\omega)\le\frac{\lVert\langle\tau,\omega\rangle\rVert+\lVert\tau\rVert\lVert\omega\rVert}{2}\le\lVert\tau\rVert\lVert\omega\rVert. \tag*{(7)}

In particular,

w_A(\tau\otimes\tau)\le\lVert\tau\rVert^2.

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

\lVert\langle x,(\tau\otimes\omega)x\rangle\rVert=\lVert\langle x,\tau\langle\omega,x\rangle\rangle\rVert=\lVert\langle x,\tau\rangle\langle\omega,x\rangle\rVert

=\lVert\langle\omega,x\rangle\langle x,\omega\rangle\rVert\le\frac{\lVert\langle\tau,\omega\rangle\rVert+\lVert\tau\rVert\lVert\omega\rVert}{2}.

Therefore

w_A(\tau\otimes\omega) \le\frac{\left\|\langle\tau,\omega\rangle\right\|+\|\tau\|\|\omega\|}{2}.

Unlike the Hilbert space case, we are unable to derive equality in Inequality (7). Let $\mathcal{A}$ be a unital C*-algebra and $p(z):=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in\mathcal{A}[z]$ . Define the modular Frobenius companion matrix

C_p:=\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& \ddots& \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 M_n(\mathcal{A}).

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

\left\langle(a_j)_{j=1}^n,(b_j)_{j=1}^n\right\rangle:=\sum_{j=1}^n a_j^*b_j,\quad\forall(a_j)_{j=1}^n,(b_j)_{j=1}^n\in\mathcal{A}^n.

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

\left\|(a_j)_{j=1}^n\right\|=\left\|\sum_{j=1}^n a_j^*a_j\right\|^{\frac{1}{2}},\quad\forall(a_j)_{j=1}^n\in\mathcal{A}^n.

We then have the following result.

Theorem 2.15. Let $\mathcal{A}$ be a commutative unital C*-algebra and $p(z):=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in\mathcal{A}[z]$ . If $b\in\mathcal{A}$ satisfies $p(b)=0$ , then $b\in W_{\mathcal{A}}(C_p)$ .

Proof. Define

x:=\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}}.

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

\begin{aligned} \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^*)^j b^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& \ddots& \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^*)^j b^j\right)^{-\frac{1}{2}} \right\rangle\\ &= \left(\sum_{j=0}^{n-1}(b^*)^j b^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& \ddots& \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^*)^j b^j\right)^{-\frac{1}{2}}\\ &= \left(\sum_{j=0}^{n-1}(b^*)^j b^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_jb^j\\ b^{n-1}\\ b^{n-2}\\ \vdots\\ b^3\\ b^2\\ b \end{pmatrix} \right\rangle \left(\sum_{j=0}^{n-1}(b^*)^j b^j\right)^{-\frac{1}{2}}\\ &= \left(\sum_{j=0}^{n-1}(b^*)^j b^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^*)^j b^j\right)^{-\frac{1}{2}}\\ &= \left(\sum_{j=0}^{n-1}(b^*)^j b^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^*)^j b^j\right)^{-\frac{1}{2}}\\ &= \left(\sum_{j=0}^{n-1}(b^*)^j b^j\right)^{-\frac{1}{2}} \left(\sum_{j=0}^{n-1}(b^*)^j b^j\right)b \left(\sum_{j=0}^{n-1}(b^*)^j b^j\right)^{-\frac{1}{2}}\\ &= \left(\sum_{j=0}^{n-1}(b^*)^j b^j\right)^{-\frac{1}{2}} \left(\sum_{j=0}^{n-1}(b^*)^j b^j\right) \left(\sum_{j=0}^{n-1}(b^*)^j b^j\right)^{-\frac{1}{2}}b\\ &=b. \tag*{(8)} \end{aligned}

Therefore $b=\langle x,C_p x\rangle\in W_A(C_p)$ .

We now derive modular analogue of Theorem 1.4.

Theorem 2.16. Let $\mathcal{A}$ be a commutative unital $C^*$ -algebra and $p(z):=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in\mathcal{A}[z]$ . If $b\in\mathcal{A}$ satisfies $p(b)=0$ , then

\lVert b\rVert\le w_{\mathcal{A}}(R)+\frac{\sqrt{\left\lVert\sum_{j=0}^{n-1}a_j^*a_j\right\rVert}+\lVert a_{n-1}\rVert}{2}

\le w_{\mathcal{A}}(R)+\frac{\sqrt{\sum_{j=0}^{n-1}\lVert a_j\rVert^2}+\lVert a_{n-1}\rVert}{2},

where $R$ is the right shift matrix defined by

R:=\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&\ddots&\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}).

Proof. Define

\mathbf{a}:=\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 \mathbf{e}_1:=\begin{pmatrix} 1\\ 0\\ 0\\ \vdots\\ 0\\ 0\\ 0 \end{pmatrix}\in\mathcal{A}^n.

Then

C_p=R-\mathbf{e}_1\otimes\mathbf{a}. \tag*{(9)}

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

\begin{aligned} \lVert b\rVert\le w_{\mathcal{A}}(C_p) =w_{\mathcal{A}}(R-\mathbf{e}_1\otimes\mathbf{a}) \le w_{\mathcal{A}}(R)+w_{\mathcal{A}}(\mathbf{e}_1\otimes\mathbf{a}) \\ &\le w_{\mathcal{A}}(R)+\frac{\sqrt{\left\lVert\sum_{j=0}^{n-1}a_j^*a_j\right\rVert}+\lVert a_{n-1}\rVert}{2} \le w_{\mathcal{A}}(R)+\frac{\sqrt{\sum_{j=0}^{n-1}\lVert a_j\rVert^2}+\lVert a_{n-1}\rVert}{2}. \end{aligned}

Let $\mathcal{A}$ be a commutative unital C $^*$ -algebra. Let $p(z):=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in\mathcal{A}[z]$ with $a_0$ invertible. Define

q(z):=\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].

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

C_q:= \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& \ddots& \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 M_n(\mathcal{A}).

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

Theorem 2.17. Let $\mathcal{A}$ be a commutative unital C $^*$ -algebra and $p(z):=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in\mathcal{A}[z]$ with $a_0$ invertible. If an invertible element $b\in\mathcal{A}$ satisfies $p(b)=0$ , then

\frac{1}{\lVert b^{-1}\rVert}\geq\frac{2}{2w_{\mathcal{A}}(R)+\sqrt{\left\lVert(a_0^{-1})^*a_0^{-1}+\sum_{j=1}^{n-1}(a_ja_0^{-1})^*a_ja_0^{-1}\right\rVert^2+\lVert a_1a_0^{-1}\rVert^2}}.

Equality term in (4) comes from the numerical radius of right shift matrix obtained by Davidson and Holbrook [25] (also see [48,59]). We are unable to do this for the right shift matrix over C $^*$ -algebras. We now derive Theorem 1.6 for Hilbert C $^*$ -modules.

Theorem 2.18. (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

w_{\mathcal{A}}(T)\leq\frac{1}{\sqrt{2}}\sqrt{w_{\mathcal{A}}(T^2)+\lVert T\rVert^2}\leq\frac{1}{\sqrt{2}}\sqrt{\lVert T^2\rVert+\lVert T\rVert^2}\leq\lVert T\rVert.

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{aligned} \lVert\langle x,Tx\rangle\rVert^2=\lVert\langle x,Tx\rangle\langle x,Tx\rangle^*\rVert=\lVert\langle x,Tx\rangle\langle Tx,x\rangle\rVert\\ &=\lVert\langle Tx,x\rangle\langle x,Tx\rangle\rVert\leq\frac{\lVert\langle Tx,x\rangle\rVert+\lVert Tx\rVert\lVert Tx\rVert}{2} \\ &=\frac{\lVert\langle x,T^2x\rangle\rVert+\lVert Tx\rVert\lVert T^*x\rVert}{2}\leq\frac{\lVert\langle x,T^2x\rangle\rVert+\lVert T\rVert\lVert x\rVert\lVert T\rVert\lVert x\rVert}{2} \\ &=\frac{\lVert\langle x,T^2x\rangle\rVert+\lVert T\rVert^2}{2}\leq\frac{\sup_{y\in\mathcal{E},\langle y,y\rangle=1}\lVert\langle y,T^2y\rangle\rVert+\lVert T\rVert^2}{2} \\ &=\frac{w_{\mathcal{A}}(T^2)+\lVert T\rVert^2}{2}. \end{aligned}

Therefore

w_{\mathcal{A}}(T)=\sup_{x\in\mathcal{E},\langle x,x\rangle=1}\lVert\langle x,Tx\rangle\rVert\leq\frac{1}{\sqrt{2}}\sqrt{w_{\mathcal{A}}(T^2)+\lVert T\rVert^2}\leq\frac{1}{\sqrt{2}}\sqrt{\lVert T^2\rVert+\lVert T\rVert^2}\leq\lVert T\rVert.

Note that we derived Theorem 2.18 only for self-adjoint morphisms. We are unable to derive Theorem 2.18 for arbitrary adjointable morphisms. We also note that in the case of self-adjoint operator $T$ on Hilbert spaces, we have $w_{\mathrm{C}}(T)=\lVert T\rVert$ [46] which we can’t say for self-adjoint morphisms on Hilbert C*-modules.

Noncommutative Toeplitz-Hausdorff Problem

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

Toeplitz-Hausdorff Theorem (fundamental theorem of the numerical range) [26, 74, 78, 85, 92, 97] [34, 45, 46, 47, 50, 51]?
Wintner spectral inclusion theorem for numerical range [8, 102] [46]?
Halmos-Bernau-Smithies-Berger power inequality [11, 82]?
Alpin-Chien-Yeh inequality [6]?
Kittaneh inequalities [63, 64]?
Yamazaki inequality [104]?
Kittaneh-Moslehian-Yamazaki inequality [66]?
Abu-Omar-Kittaneh inequality [3]?
Bhunia-Bag-Paul inequality [12]?
elliptical range theorem [17, 70]?
Berger power dilation for numerical contraction [9, 10, 95] [46]?
Ando theorem for numerical contraction [7]?
Furuta-Nakamoto theorem for numerical contractions [37]?
El-Haddad-Kittaneh numerical radius inequality [33]?
description of numerical range of $3 \times3$ matrices [60]?
description of numerical range of $4 \times4$ matrices [39]?
Holbrook numerical radius inequalities for the product of commuting matrices [54]?
Okubo-Ando numerical radius inequalities for the product of commuting matrices [77, 80] [55]?
Bouldin numerical radius inequalities for the product of commuting matrices [14, 15]?
Poncelet property for numerical ranges [4, 103] [41]?
Anderson theorem [31, 96] [44]?
Kippenhahn boundary generating curve theorem [35, 62]?
Kippenhahn corner-point theorem [35, 62]?
Hildebrandt normal eigenvalue theorem [16] [53]?
Hildebrandt intersection theorem [101] [38]?
Marcus-Shure description for the numerical range of zero-one matrices [72] and Davidson-Holbrook estimate for the numerical radius of zero-one matrices [25]?
Johnson numerical range inclusion theorem [57]?
Choi-Li constrained numerical range theorem [20]?
Wang-Wu-Gau result for Crawford number [100]?
Abu-Omar-Kittaneh inequality (involving generalized Aluthge transform) [1]?
Gau-Wu numerical radius equality characterizations [43]?
Gau-Wu theorem for compact operators [42]?
Perron-Frobenius type results on the numerical range [73]?

(34) Helton-Spitkovsky characterization of shapes of numerical ranges [52]?

(35) Radjabalipour-Radjavi result on numerical ranges [84]?

(36) Pollack theorem on numerical ranges [83]?

(37) Anderson result on numerical ranges [5]?

(38) Abu-Omar-Kittaneh numerical radius inequalities for the product of matrices [2]?

(39) Kittaneh-Moradi inequality [65]?

(40) Lancaster characterization for the closedness of numerical range [67]?

(41) Gau-Wu characterization for numerical ranges of completely non-unitary contractions [40]?

(42) Mees-Atherton domains containing numerical ranges [75]?

(43) Chien-Tam characterizations of circularity of numerical ranges [19]?

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

Noncommutative Crouzeix Inequality, von Neumann inequality and Ando inequality Problems

Breakthrough Crouzeix theorem says following.

Theorem 4.1. [13, 22, 23, 24, 87] (Crouzeix Theorem) For every $d \in\mathbb{N}$ and for every matrix $M \in\mathrm{M}_{d}(\mathbb{C})$ , we have

(Crouzeix-Palencia Inequality)

\lVert p(M)\rVert\le(1+\sqrt{2})\sup\{\lvert p(z)\rvert: z \in W_{\mathbb{C}}(M)\}, \quad\forall p \in\mathbb{C}[z].

We formulate following problem.

Problem 4.2. (Noncommutative Crouzeix Problem) Let $A$ be a unital $C^{*}$ -algebra. Whether there is a universal constant $R_{A}$ (which may depend upon $A$ ) satisfying following: For every $d \in\mathbb{N}$ and for every matrix $M \in\mathrm{M}_{d}(A)$ , we have

\lVert p(M)\rVert\le R_{A}\sup\{\lVert p(z)\rVert: z \in W_{A}(M)\}, \quad\forall p \in A[z].

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

Theorem 4.3. [27, 79, 94, 95, 99] (von Neumann Theorem) For every $d \in\mathbb{N}$ and for every matrix $M \in\mathrm{M}_{d}(\mathbb{C})$ with $\lVert M\rVert\le1$ , we have

(von Neumann Inequality)

\lVert p(M)\rVert\le\sup\{\lvert p(z)\rvert: z \in\mathbb{C}, \lvert z\rvert\le1\}, \quad\forall p \in\mathbb{C}[z].

Based on Theorem 4.3 we formulate following problems.

Problem 4.4. (Noncommutative von Neumann Inequality Problem) Let $A$ be a unital $C^{*}$ -algebra. Whether there is a universal constant $R_{A}$ (which may depend upon $A$ ) satisfying following: For every $d \in\mathbb{N}$ and for every $A \in\mathrm{M}_{d}(A)$ with $\lVert A\rVert\le1$ , we have

\lVert p(A)\rVert\le R_{A}\sup\{\lVert p(z)\rVert: z \in A, \lVert z\rVert\le1\}, \quad\forall p \in A[z].

Theorem 4.3 has been extended by Ando for two variables.

Theorem 4.5. [7, 95] (Ando Theorem) For every $d \in\mathbb{N}$ and for all matrices $M,N \in\mathrm{M}_{d}(\mathbb{C})$ with $\lVert M\rVert\le1$ , $\lVert N\rVert\le1$ and $MN=NM$ , we have

(Ando Inequality)

\lVert p(M,N)\rVert\le\sup\{\lvert p(z,w)\rvert: z,w \in\mathbb{C}, \lvert z\rvert\le1, \lvert w\rvert\le1\}, \quad\forall p \in\mathbb{C}[z,w].

Based on Theorem 4.5, we formulate following problem.

Problem 4.6. (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 $A,B \in M_d(\mathcal{A})$ with $\lVert A\rVert\le1$ , $\lVert B\rVert\le1$ and $AB=BA$ , we have

\lVert p(A,B)\rVert\le\sup\{\lVert p(z,w)\rVert: z,w \in\mathcal{A},\lVert z\rVert\le1,\lVert w\rVert\le1\},\qquad\forall p \in\mathcal{A}[z,w].

Note that Ando theorem cannot be extended to more than two commuting matrices [21, 98]. We observe that the Halmos dilation [49], Egervary dilation [32, 69] and Sz.-Nagy dilation (Schaffer construction) [89] carry over to adjointable morphisms on Hilbert $C^*$ -modules. This observation will give following result.

Theorem 4.7. (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 $\lVert T\rVert\le1$ . 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

T^n x=P_{\mathcal{E}}U^n x,\qquad\forall n\in\mathbb{N},\forall x\in\mathcal{E}.

Corollary 4.8. 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 $\lVert T\rVert\le1$ . Let $\mathcal{E}_0$ , $P_{\mathcal{E}}$ and $U$ be as in Theorem 4.7. Then

\lVert p(T)\rVert\le\lVert p(U)\rVert,\qquad\forall p(z)=a_0+za_1+\cdots+z^n a_n\in\mathcal{A}[z].

Corollary 4.9. 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 $\lVert T\rVert\le1$ . 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 4.7. We then have

x=Tx=P_{\mathcal{E}}Ux,\qquad\langle Ux,Ux\rangle=\langle x,x\rangle,\qquad P_{\mathcal{E}}x=x.

We see that

\begin{aligned} \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{aligned}

Therefore

Ux=x\Longrightarrow x=U^*x.

Hence

x=U^*x=P_{\mathcal{E}}U^*x=T^*x.

We also observe that the proof of Ando dilation [7] 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).

Acknowledgments

AI is used for the literature survey and grammar improvement.

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Noncommutative Buzano-Dragomir Inequality and Applications

Abstract

Introduction

Noncommutative Buzano-Dragomir Inequality

Noncommutative Toeplitz-Hausdorff Problem

Noncommutative Crouzeix Inequality, von Neumann inequality and Ando inequality Problems

Acknowledgments

References

Publication

Taxonomy

Authors

AI contributions

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