Numerical radius inequalities of operator matrices with applications

TL;DR

This paper develops improved bounds for the numerical radius of 2x2 operator matrices and applies these results to obtain better estimates for polynomial zeros.

Contribution

It introduces tighter bounds for the numerical radius of operator matrices and demonstrates their application in polynomial zero estimation.

Findings

Improved bounds for the numerical radius of 2x2 operator matrices.

Enhanced estimation of polynomial zeros using the new bounds.

Better understanding of operator matrix behavior in numerical radius context.

Abstract

We present upper and lower bounds for the numerical radius of $2 \times 2$ operator matrices which improves on the existing bound for the same. As an application of the results obtained we give a better estimation for the zeros of a polynomial.

Equations

$$\frac{1}{2}\Vert T\Vert \le w(T)\le \Vert T\Vert .$$

$$\frac{1}{4}\Vert T^{\ast }T+T{T}^{\ast }\Vert \le w^2(T)\le \frac{1}{2}\Vert T^{\ast }T+T{T}^{\ast }\Vert$$

$$w(T)\le \frac{1}{2}(\Vert T\Vert +\Vert T^2{\Vert }^{\frac{1}{2}}).$$

$$\frac{1}{2}\sqrt{\Vert T^{\ast }T+T{T}^{\ast }\Vert +2m(T^2)}$$

$$w^{4}\left(\begin{array}[]{cc}0&X\\ Y&0\end{array}\right)\leq\frac{1}{16}\|S\|^{2}+\frac{1}{4}w^{2}(YX)+\frac{1}{8}w(YXS+SYX),$$

$$f(\theta )$$

$$\Rightarrow f^4(\theta )$$

$$w^{4}\left(\begin{array}[]{cc}0&X\\ Y&0\end{array}\right)\leq\frac{1}{16}\|S\|^{2}+\frac{1}{4}w^{2}(YX)+\frac{1}{8}w(YXS+SYX).$$

$$w^{4}\left(\begin{array}[]{cc}0&X\\ Y&0\end{array}\right)\leq\frac{1}{16}\|P\|^{2}+\frac{1}{4}w^{2}(XY)+\frac{1}{8}w(XYP+PXY),$$

$$w(XY)\le \frac{1}{4}\sqrt{\Vert S{\Vert }^2+4w^2(YX)+2w(YXS+SYX)}$$

$$w(XY)$$

$$w^{4}\left(\begin{array}[]{cc}0&X\\ Y&0\end{array}\right)\geq\frac{1}{16}\|S\|^{2}+\frac{1}{4}c^{2}(YX)+\frac{1}{8}m(YXS+SYX),$$

$$\displaystyle w\left(\begin{array}[]{cc}0&X\\ Y&0\end{array}\right)$$

$$w^{4}\left(\begin{array}[]{cc}0&X\\ Y&0\end{array}\right)\geq\frac{1}{16}\|S\|^{2}+\frac{1}{4}c^{2}(YX)+\frac{1}{8}m(YXS+SYX).$$

$$w^{4}\left(\begin{array}[]{cc}0&X\\ Y&0\end{array}\right)\geq\frac{1}{16}\|P\|^{2}+\frac{1}{4}c^{2}(XY)+\frac{1}{8}m(XYP+PXY),$$

$$w\left(\begin{array}[]{cc}X&Y\\ Z&W\end{array}\right)\geq w\left(\begin{array}[]{cc}0&Y\\ Z&0\end{array}\right)$$

$$w\left(\begin{array}[]{cc}X&Y\\ Z&W\end{array}\right)\geq w\left(\begin{array}[]{cc}X&0\\ 0&W\end{array}\right).$$

$$w\left(\begin{array}[]{cc}X&Y\\ Z&W\end{array}\right)\leq\max\{w(X),w(W)\}+[\frac{1}{16}\|S\|^{2}+\frac{1}{4}w^{2}(ZY)+\frac{1}{8}w(ZYS+SZY)]^{\frac{1}{4}}$$

$$w\left(\begin{array}[]{cc}X&Y\\ Z&W\end{array}\right)\geq\max\{w(X),w(W),[\frac{1}{16}\|S\|^{2}+\frac{1}{4}c^{2}(ZY)+\frac{1}{8}m(ZYS+SZY)]^{\frac{1}{4}}\},$$

$$C(p)$$

$$|\lambda |\le 1+\max \{|a_0|,|a_1|,\dots ,|a_{n-1}|\}.$$

$$|\lambda |\le (1+|a_0{|}^2+|a_1{|}^2+\dots +|a_{n-1}{|}^2{)}^{\frac{1}{2}}.$$

$$|\lambda |\le \max \{1,|a_0|+|a_1|+\dots +|a_{n-1}|\}.$$

$$|\lambda|\leq\cos\frac{\pi}{n+1}+\frac{1}{2}\big{[}\big{(}\sum_{j=0}^{n-1}|a_{j}|^{2}\big{)}^{\frac{1}{2}}+|a_{n-1}|\big{]}.$$

$$|\lambda |\le \underset{1\le k\le n}{\max }[(1+|a_{n-1}|)(1+|a_{n-2}|)\dots (1+|a_{n-k}|){]}^{\frac{1}{k}}.$$

$$|\lambda|\leq\frac{1}{2}\big{[}w(A)+\cos\frac{\pi}{n-1}+\sqrt{(w(A)-\cos\frac{\pi}{n-1})^{2}+(1+\sqrt{\sum_{k=3}^{n}|a_{n-k}|^{2}})^{2}}\big{]},$$

$$|\lambda|\leq\frac{1}{2}\big{[}\frac{1}{2}(|a_{n-1}|+\alpha)+\cos\frac{\pi}{n+1}+\sqrt{(\frac{1}{2}(|a_{n-1}|+\alpha)-\cos\frac{\pi}{n+1})^{2}+4\alpha^{\prime}}\big{]},$$

$$\lambda\leq\max\{w(A),\cos\frac{\pi}{n+1}\}+\frac{1}{2}\big{(}1+\sqrt{{\sum_{j=0}^{n-3}|a_{j}|^{2}}}\big{)},$$

$$|\lambda |$$

$${\alpha }_r$$

$$\alpha$$

Full text

The first author would like to thank UGC, Govt. of India for the financial support in the form of junior research fellowship.

1. Introduction

Let $B(H)$ denote the $C^{*}$-algebra of all bounded linear operators on a complex Hilbert space $H$ with usual inner product $\langle.,.\rangle$. Let $H_{1},H_{2}$ be two Hilbert spaces and $B(H_{1},H_{2})$ be the set of all bounded linear operators from $H_{1}$ into $H_{2}$. If $H_{1}=H_{2}=H$ then we write $B(H_{1},H_{2})=B(H)$. For $T\in B(H)$, the operator norm of $T$, denoted as $\|T\|$, is defined as $\|T\|=\sup\{\|Tx\|:x\in H,~{}\|x\|=1\}$. The numerical range of $T$, denoted as $W(T),$ is defined as $W(T)=\{\langle Tx,x\rangle:x\in H,~{}\|x\|=1\}$. The spectrum of $T$, denoted as $\sigma(T)$ is defined as the collection of all spectral values of $T.$ The numerical radius and spectral radius of $T$, denoted as $w(T)$ and $\rho(T)$ respectively, are defined as the radius of the smallest circle with centre at origin which contains the numerical range and spectrum of $T$. It is well known that $\sigma(T)\subseteq\overline{W(T)}$ and so $\rho(T)\leq w(T).$ The Crawford number of $T$, denoted as $m(T),$ is defined as $m(T)=\inf\{|\langle Tx,x\rangle|:x\in H,\|x\|=1\}.$ It is easy to see that $w(T)$ is a norm on $B(H)$, equivalent to the operator norm that satisfies the inequality

[TABLE]

The first inequality becomes an equality if $T^{2}=0$ and the second inequality becomes an equality if $T$ is normal. Various numerical radius inequalities improving this inequality have been studied in [9, 12, 14, 15, 16]. Kittaneh in [11, 12] respectively, proved that if $T\in B(H)$ then

[TABLE]

and

[TABLE]

Also Abu-Omar and Kittaneh in [2] proved that if $T\in B(H)$ then

[TABLE]

They also proved that the upper bound of $w(T)$ in this inequality is better than the above upper bounds in [11, 12].

Here we present an upper bound for the numerical radius of $2\times 2$ operator matrices which improves the existing bound in [1]. Also we present lower bounds for the numerical radius of $2\times 2$ operator matrices. As an application of the results obtained we estimate bounds for the zeros of a complex polynomial. Also, we show with numerical examples that the bounds obtained by us for the zeros of a polynomial improves on the existing bounds.

2. Main results

We begin this section with the following lemmas which are used to reach our goal in this present article. These four lemmas can be found in [1, 3, 8].

Lemma 2.1** ([1]).**

Let $T\in B(H)$, then $w(T)=\sup_{\theta\in\mathbb{R}}\|\textit{Re}(e^{i\theta}T)\|$.

Lemma 2.2** ([1]).**

*Let $X\in B(H_{1}),Y\in B(H_{2},H_{1}),Z\in B(H_{1},H_{2})$ and $W\in B(H_{2})$. Then the following results hold:

$(i)$ w\left(\begin{array}[]{cc}X&0\\ 0&W\end{array}\right)=\max\{w(X),w(W)\}.

$(ii)$ w\left(\begin{array}[]{cc}0&Y\\ Z&0\end{array}\right)=w\left(\begin{array}[]{cc}0&Z\\ Y&0\end{array}\right).

$(iii)$ w\left(\begin{array}[]{cc}0&Y\\ Z&0\end{array}\right)=\sup_{\theta\in\mathbb{R}}\frac{1}{2}\|e^{i\theta}Y+e^{-i\theta}Z^{*}\|.

$(iv)$ If   $H_{1}=H_{2}$, then   w\left(\begin{array}[]{cc}0&Y\\ Y&0\end{array}\right)=w(Y).*

Lemma 2.3** ([8]).**

Let $C,T\in B(H)$. Then $w(TC+C^{*}T)\leq 2w(T)$, where $C$ is any contraction $(\textit{i.e.},\|C\|\leq 1)$.

Lemma 2.4** ([3]).**

If D=\left(\begin{array}[]{cccccc}0&0&.&.&.&0\\ 1&0&.&.&.&0\\ .&&&&&\\ .&&&&&\\ .&&&&&\\ 0&0&.&.&1&0\end{array}\right)_{n,n} then $w(D)=\cos\frac{\pi}{n+1}$.

Now we are ready to prove the following inequality for the numerical radius of $2\times 2$ operator matrices which improves on the existing inequalities.

Theorem 2.5**.**

Let $X\in B(H_{2},H_{1}),Y\in B(H_{1},H_{2})$. Then

[TABLE]

where $S=|X|^{2}+|Y^{*}|^{2}$.

Proof.

Let $f(\theta)=\frac{1}{2}\|e^{i\theta}X+e^{-i\theta}Y^{*}\|$. Therefore,

[TABLE]

Now taking supremum over $\theta\in\mathbb{R}$ in the above inequality and then from Lemma 2.2 $(iii)$ and Lemma 2.1 we get,

[TABLE]

This completes the proof. ∎

Now using Lemma 2.2 $(ii)$ and Theorem 2.5 we get the following inequality.

Corollary 2.6**.**

Let $X\in B(H_{2},H_{1}),Y\in B(H_{1},H_{2})$. Then

[TABLE]

where $P=|X^{*}|^{2}+|Y|^{2}$.

Again using Lemma 2.2 $(i)$ and Theorem 2.5 we get the following inequality.

Corollary 2.7**.**

Let $X\in B(H_{2},H_{1}),Y\in B(H_{1},H_{2})$. Then

[TABLE]

where $S=|X|^{2}+|Y^{*}|^{2}$.

Proof.

Now,

[TABLE]

∎

Remark 2.8**.**

Using Lemma 2.3, it is easy to observe that the bound obtained in Theorem 2.5 is better than the second inequality in [1, Th. 3].

Remark 2.9**.**

Here we note that when $H_{1}=H_{2}$ and $Y=X$ then it follows from Theorem 2.5 and Lemma 2.2 $(iv)$ that $w^{4}(X)\leq\frac{1}{16}\|R\|^{2}+\frac{1}{4}w^{2}(X^{2})+\frac{1}{8}w(X^{2}R+RX^{2}),$ where $R=|X|^{2}+|X^{*}|^{2}$. This inequality can be found in [6, Th. 2.1]

Next we prove a lower bound for the numerical radius of $2\times 2$ operator matrices.

Theorem 2.10**.**

Let $X\in B(H_{2},H_{1}),Y\in B(H_{1},H_{2})$. Then

[TABLE]

where $S=|X|^{2}+|Y^{*}|^{2},~{}c(YX)=\inf_{\theta\in\mathbb{R}}\inf_{x\in H_{2},\|x\|=1}\|\textit{Re}(e^{i\theta}YX)x\|$.

Proof.

Let $x\in H_{2}$ with $\|x\|=1$ and $\theta$ be a real number such that $e^{2i\theta}\langle(YXS+SYX)x,x\rangle=|\langle(YXS+SYX)x,x\rangle|$. Then from Lemma 2.2 $(iii)$ we get,

[TABLE]

Now taking supremum over $x\in H_{2}$ with $\|x\|=1$ in the above inequality we get,

[TABLE]

This completes the proof. ∎

Now using Lemma 2.2$(ii)$ and Theorem 2.10 we get the following inequality.

Corollary 2.11**.**

Let $X\in B(H_{2},H_{1}),Y\in B(H_{1},H_{2})$. Then

[TABLE]

where $P=|X^{*}|^{2}+|Y|^{2}$.

Remark 2.12**.**

Here we note that when $H_{1}=H_{2}$ and $Y=X$ then it follows from Theorem 2.10 and Lemma 2.2 $(iv)$ that $w^{4}(X)\geq\frac{1}{16}\|R\|^{2}+\frac{1}{4}c^{2}(X^{2})+\frac{1}{8}m(X^{2}R+RX^{2}),$ where $R=|X|^{2}+|X^{*}|^{2}$. Also, this inequality can be found in [6, Th. 3.1]

Next we state the following lemma which can be found in [9].

Lemma 2.13**.**

Let $X,Y,Z,W\in B(H)$. Then

[TABLE]

and

[TABLE]

Now we are ready to prove an upper bound and a lower bound for the numerical radius of an operator matrix \left(\begin{array}[]{cc}X&Y\\ Z&W\end{array}\right), where $X,Y,Z,W\in B(H)$.

Theorem 2.14**.**

Let $X,Y,Z,W\in B(H)$. Then

[TABLE]

and

[TABLE]

where $S=|Y|^{2}+|Z^{*}|^{2},c(ZY)=\inf_{\theta\in\mathbb{R}}\inf_{x\in H,\|x\|=1}\|\textit{Re}(e^{i\theta}ZY)x\|$.

Proof.

The proof follows easily from the Theorem 2.5, Theorem 2.10 and Lemma 2.13. ∎

3. Application

Let us consider a monic polynomial $p(z)=z^{n}+a_{n-1}z^{n-1}+\ldots+a_{1}z+a_{0}$ with complex coefficients $a_{0},a_{1},\ldots,a_{n-1}$. Then the Frobenius companion matrix of $p(z)$ is given by

[TABLE]

Then the zeros of the polynomial $p(z)$ are exactly the eigenvalues of $C(p)$. Also we know that the spectrum $\sigma(C(p))\subseteq\overline{W(C(p))}$, so that if $z$ is a zero of the polynomial $p(z)$ then $|z|\leq w(C(p))$.

Many mathematicians have estimated the zeros of the polynomial using this above argument, some of them are mentioned below. Let $\lambda$ be a zero of the polynomial $p(z)$.

(1) Cauchy [10] proved that

[TABLE]

(2) Carmichael and Mason [10] proved that

[TABLE]

(3) Montel [10] proved that

[TABLE]

(4) Fujii and Kubo [7] proved that

[TABLE]

(5) Alpin et. al. [5] proved that

[TABLE]

(6) Paul and Bag [13] proved that

[TABLE]

where A=\left(\begin{array}[]{cc}-a_{n-1}&-a_{n-2}\\ 1&0\end{array}\right).

(7) Abu-Omar and Kittaneh [3] proved that

[TABLE]

where $\alpha=\sqrt{\sum_{j=0}^{n-1}|a_{j}|^{2}}$ and $\alpha^{\prime}=\sqrt{\sum_{j=0}^{n-2}|a_{j}|^{2}}$.

(8) M. Al-Dolat et. al. [4] proved that

[TABLE]

where A=\left(\begin{array}[]{cc}-a_{n-1}&-a_{n-2}\\ 1&0\end{array}\right).

We first prove the following theorem.

Theorem 3.1**.**

Let $\lambda$ be any zero of $p(z)$. Then

[TABLE]

where

[TABLE]

Proof.

Putting $z=\eta-\frac{a_{n-1}}{n}$ in the polynomial $p(z)$ we get, a polynomial

$q(\eta)=\eta^{n}+\alpha_{n-2}\eta^{n-2}+\alpha_{n-3}\eta^{n-3}+\ldots+\alpha_{1}\eta+\alpha_{0},$

where \alpha_{r}=\sum^{n}_{k=r}{{}^{k}}C_{r}\big{(}-\frac{a_{n-1}}{n}\big{)}^{k-r}a_{k},~{}~{}r=0,1,\ldots,n-2, $a_{n}=1$ and ${}^{0}C_{0}=1$.

Now the Frobenius companion matrix of the polynomial $q(\eta)$ is C(q)=\left(\begin{array}[]{cc}A&B\\ C&D\end{array}\right) where $A=(0)_{1,1}$, $B=(-\alpha_{n-2}~{}~{}-\alpha_{n-3}~{}~{}\ldots~{}~{}-\alpha_{1}~{}~{}-\alpha_{0})_{1,n-1}$, $C^{t}=(1~{}~{}0~{}~{}\ldots~{}~{}0~{}~{}0)_{1,n-1}$, D=\left(\begin{array}[]{cccccc}0&0&.&.&.&0\\ 1&0&.&.&.&0\\ .&&&&&\\ .&&&&&\\ .&&&&&\\ 0&0&.&.&1&0\end{array}\right)_{n-1,n-1}.

Now using Lemma 2.2$(i)$ and Lemma 2.4 we get,

[TABLE]

Therefore, if $\eta$ is any zero of the polynomial $q(\eta)$ then |\eta|\leq\cos\frac{\pi}{n}+w\left(\begin{array}[]{cc}0&B\\ C&0\end{array}\right). Therefore if $\lambda$ is any zero of the polynomial $p(z)$ then |\lambda|\leq|\frac{a_{n-1}}{n}|+\cos\frac{\pi}{n}+w\left(\begin{array}[]{cc}0&B\\ C&0\end{array}\right). Now using the Theorem 2.5 in the above inequality we get,

[TABLE]

This completes the proof of the theorem. ∎

Remark 3.2**.**

There is an error in the summation formula for computation of the coefficient $b_{r}$ in [13, page 237] and $\alpha_{r}$ in [14, Th. 3.1].

We illustrate with numerical examples to show that the above bound obtained by us in Theorem 3.1 is better than the existing bounds.

Example 3.3**.**

Consider the polynomial $p(z)=z^{5}+2z^{4}+z+1$. Then the upper bounds of the zeros of this polynomial $p(z)$ estimated by different mathematicians are as shown in the following table.

[TABLE]

But our bound obtained in Theorem 3.1 gives $|\lambda|\leq 2.625$ which is better than all the estimations mentioned above.