An estimate for the numerical radius of the Hilbert space operators and a numerical radius inequality

1.   Introduction

Let $\mathcal{H}$ be complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the $C^*$-algebra of all bounded linear operator on $\mathcal{H}$. An operator ${\bf T}\in \mathcal{B}(\mathcal{H})$ is said to be positive if $\left < { {\bf T}\xi, \xi}\right > \geq 0$ holds for all $\xi\in \mathcal{H}$. We write ${\bf T}\geq 0$ if ${\bf T}$ is positive.

The numerical radius of ${\bf T}\in \mathcal{B}(\mathcal{H})$ is defined by

It is well known that $w(\cdot)$ defines a norm on $\mathcal{B}(\mathcal{H})$, which is equivalent to the usual operator norm $\left\Vert{\cdot}\right\Vert$. In fact, for any ${\bf T}\in \mathcal{B}(\mathcal{H})$,

Also, if ${\bf T}\in \mathcal{B}(\mathcal{H})$ is normal, then $w({\bf T}) = \left\Vert{ {\bf T}}\right\Vert$.

An important inequality for $w({\bf T})$ is the power inequality stating that $w({\bf T}^n)\leq (w({\bf T}))^n$ for every natural numbers $n$.

Several numerical radius inequalities improving the inequalities in (1.1) have been recently given in ^{[5,6,12,19,20,21,22]}. For instance, Kittaneh ^[13,14] proved that for any ${\bf A}\in \mathcal{B}(\mathcal{H})$,

where $| {\bf A}| = \sqrt{ {\bf A}^* {\bf A}}$ is the absolute value of ${\bf A}$, and

Also, in the same paper, it was shown that

Kittaneh and El-Haddad ^[15] established the generalizations of inequality (1.2) and the second inequality (1.3) as follows:

and

where $0 < \lambda < 1$ and $r\geq 1$.

A general numerical radius inequality has been established by Kittaneh ^[14], it has been proved that if ${\bf A}, {\bf B}, {\bf C}, {\bf D}, {\bf T}, {\bf S}\in \mathcal{B}(\mathcal{H})$, then

for all $\alpha\in(0, 1)$.

Although several open problems relating to numerical radius inequalities for bounded linear operators remain unsolved, work on establishing numerical radius inequalities for a number of bounded linear operators has begun (see, for example, ^[10] and ^{[19,20,21,22]}). If ${\bf A}, {\bf B}\in \mathcal{B}(\mathcal{H})$, then

In the case that ${\bf A} {\bf B} = {\bf B} {\bf A}$, we have

Moreover, if ${\bf A}$ and ${\bf B}$ are normal, then

Recently, Dragomir ^[7] proved that if ${\bf A}, {\bf B}\in \mathcal{B}(\mathcal{H})$ and $r\geq 1$, then

Shebrawi and Albadawi ^[23] discovered a fascinating numerical radius inequality, it has been shown that if ${\bf A}, {\bf X}, {\bf B} \in \mathcal{B}(\mathcal{H})$, then

Recently, Al-Dolat and Al-Zoubi ^[3], showed that if ${\bf A}_i, {\bf B}_i, {\bf X}_i\in \mathcal{B}(\mathcal{H})$ ($i = 1, 2, \cdots, n$), $m\in \mathbb N$ and $\phi$ and $\psi$ are non-negative functions on $[0, \infty)$ which are continuous such that $\phi(t)\psi(t) = t$ for all $t \in [0, \infty)$, then

for $r\geq 1$, where $E_{i, j} = {\bf X}_i \phi^2\left({\left\vert{ {\bf A}_i^{j*}}\right\vert}\right) {\bf X}_i^*$ and $W_{i, j} = ({\bf A}_i^{m-j} {\bf B}_i)^*\psi^2\left({\left\vert{ {\bf A}_i^j}\right\vert}\right) {\bf A}_i^{m-j} {\bf B}_i$.

The goal of this study is to develop significant extensions of these inequalities based on the classic convexity inequalities for nonnegative real numbers and some operator inequalities. For the sum of two operators, usual operator norm inequalities and a related numerical radius inequality are also provided. In specifically, if $i = 1, 2, \cdots n \in \mathbb N$, ${\bf A}_i, {\bf B}_i$, and ${\bf X}_i$ are bounded linear operators, then we estimate the numerical radius to $\sum_{i = 1}^{m} {\bf X}_i {\bf A}_i^m {\bf B}_i$ for some $m \in \mathbb N$.

2.   Inequalities for sums and products of operators

Using well-known new numerical radius inequalities as an example, we constructed a general numerical radius inequality for Hilbert space operators in this section. This section is initiated with an operator for norm inequality. In fact, we provide an extra upper bound for $\left\Vert{ {\bf B}^* {\bf A}+ {\bf D}^* {\bf C}}\right\Vert$. However, the proof of the theorem depends on the next lemma.

Lemma 2.1 (^[4]). Let $\xi, \zeta, \eta\in \mathcal{H}$. Then we have

Theorem 2.2. Let ${\bf A}, {\bf B}, {\bf C}, {\bf D}\in \mathcal{B}(\mathcal{H})$. Then

Proof. For $\xi, \zeta\in \mathcal{H}$, we have by triangle inequality, we have

Now in inequality (2.1), for all $\xi, \zeta\in \mathcal{H}$, letting $\xi = {\bf A}^* {\bf B}\zeta$, $\zeta = {\bf C} {\bf D}^*\zeta$ and $\eta = \xi$ with $\left\Vert{\xi}\right\Vert = \left\Vert{\zeta}\right\Vert = 1$, we get

Now use the fact that

we have

Combining the inequalities (2.2) and (2.3), we have

Taking the supremum over all unit vectors $\xi, \zeta$, we obtain the desired inequality. □

In Theorem 2.2, if we let ${\bf A} = {\bf B} = {\bf C} = {\bf D} = {\bf S}$, we have:

Corollary 2.3. Let ${\bf S}\in \mathcal{B}(\mathcal{H})$. Then

In the proof of Theorem 2.2, if we let $\xi = \zeta$, we have:

Corollary 2.4. Let ${\bf A}, {\bf B}, {\bf C}, {\bf D}\in \mathcal{B}(\mathcal{H})$. Then

The following lemma gives a basic but useful extension for four operators of the Schwarz inequality due to Dragomir ^[8].

Lemma 2.5. Let ${\bf A}, {\bf B}, {\bf C}, {\bf D}\in \mathcal{B}(\mathcal{H})$. Then for $\xi, \zeta\in \mathcal{H}$ we have the inequality

The equality case holds if and only if the vectors ${\bf B} {\bf A}\xi$ and ${\bf C}^* {\bf D}^*\zeta$ are linearly dependent in $\mathcal{H}$.

The following lemma, known as the Hölder-McCarthy inequality, is a well-known conclusion derived from Jensen's inequality and the spectral theorem for positive operators (see ^[12]).

Lemma 2.6. Let ${\bf T}\in \mathcal{B}(\mathcal{H})$, ${\bf T}\geq 0$ and let $\xi\in \mathcal{H}$ be any unit vector. Then we have

(i) $\left < { {\bf T}\xi, \xi}\right > ^r\leq \left < { {\bf T}^r\xi, \xi}\right >$ for $r\geq 1$.

(ii) $\left < { {\bf T}^r\xi, \xi}\right > \leq \left < { {\bf T}\xi, \xi}\right > ^r$ for $0 < r\leq 1$.

(iii) If ${\bf T}$ is invertible, then $\left < { {\bf T}\xi, \xi}\right > ^r\leq \left < { {\bf T}^r\xi, \xi}\right >$ for all $r < 0$.

The next result is well known in the literature as the Mond-Pečarić inequality ^[18].

Lemma 2.7. If $\psi$ is a convex function on a real interval $J$ containing the spectrum of the self-adjoint operator ${\bf T}$, then for any unit vector $\xi\in \mathcal{H}$,

and the reverse inequality holds if $\psi$ is concave.

The forth lemma is a direct consequence of [2, Theorem 2.3].

Lemma 2.8. Let $\psi$ be a non-negative non-decreasing convex function on $[0, \infty)$ and let ${\bf T}, {\bf S}\in \mathcal{B}(\mathcal{H})$ be positive operators. Then for any $0 < \mu < 1$,

The above four lemmas admit the following more general result.

Theorem 2.9. Let ${\bf A}, {\bf B}, {\bf C}, {\bf D}\in \mathcal{B}(\mathcal{H})$. If $\psi$ is a non-negative increasing convex function on $[0, \infty)$, then for any $0 < \mu < 1$,

In particular,

for all $r\geq 1$.

Proof. For any unit vector $\xi\in \mathcal{H}$, we have

Taking the supremum over $\xi\in \mathcal{H}$ with $\left\Vert{\xi}\right\Vert = 1$, we infer that

On account of assumptions on $\psi$, we can write

The inequality (2.7) follows directly from (2.6) by taking $\psi(t) = t^r$ ($r\geq 1$). □

In the following result, we want to improve (1.9) under certain mild situations. We'll need the arithmetic-geometric mean inequality refinement ^[24] to do this.

Lemma 2.10. Suppose that $\mu, \nu > 0$ and positive real numbers $\delta, \Delta$ satisfy

Then

The following lemma is very useful in the proof of the next result.

Lemma 2.11. Let $\psi$ be a non-negative increasing convex function on $[0, \infty)$, $\psi(0) = 0$ and $\alpha\in [0, 1]$. Then $\psi(\alpha t)\leq \alpha \psi(t)$.

Theorem 2.12. Let ${\bf A}, {\bf B}, {\bf C}, {\bf D}\in \mathcal{B}(\mathcal{H})$ and let $\psi$ be a non-negative increasing convex function on $[0, \infty)$. If

or

then

Proof. It follows from Lemma 2.5 that

Combining (2.10) and (2.11), we obtain

Taking the supremum over $\xi\in \mathcal{H}$ with $\left\Vert{\xi}\right\Vert = 1$, we infer that

Now, since $\psi$ is a non-negative increasing convex function, we have

□

As an applications of Theorem 2.12, we have:

Corollary 2.13. Let ${\bf T}\in \mathcal{B}(\mathcal{H})$, $\alpha, \beta\geq 0$ with $\alpha+\beta\geq 1$ and let $\psi$ be a non-negative increasing convex function on $[0, \infty)$. If

or

then

Remark 2.14. Following (2.9) we list here some particular inequalities of interest.

$({\rm{i}})$ If we let $\psi(t) = t^r$ ($r\geq 1$), we have

whenever

$({\rm{ii}})$ Letting ${\bf D} = {\bf S}^*, {\bf A} = {\bf T}$ and let $\psi(t) = t^r$ ($r\geq 1$), we have

whenever

$({\rm{iii}})$ Letting ${\bf C} = {\bf D} = {\bf B} = I$ and ${\bf A} = {\bf T}$ and let $\psi(t) = t^r$ ($r\geq 1$), we have

whenever

We give an example to clarify part (ⅱ) in Remark 2.14

Example 2.15. Let ${\bf S} = \begin{bmatrix}3/2 & 1/2 \\1/2 & 3/2 \\ \end{bmatrix}$ and ${\bf T} = \begin{bmatrix}1/2 & 0 \\0 & 1/2 \\ \end{bmatrix}$ and $r = 2$. A simple calculations show that ${\bf S}^* {\bf T} = \begin{pmatrix}\frac{3}{4} & \frac{1}{4}\\ \frac{1}{4} & \frac{3}{4}\end{pmatrix}$ and so $w^2({\bf S}^* {\bf T}) = 1$, $\left\Vert{| {\bf S}|^4+| {\bf T}|^4}\right\Vert = \frac{257}{16} = 16.0625$. If we take $\delta = 0.3$ and $\Delta = .4$, then

If ${\bf A}, {\bf B}\in \mathcal{B}(\mathcal{H})$ are positive, the geometric mean of ${\bf A}$ and ${\bf B}$, denoted by ${\bf A}\sharp {\bf B}$, is defined as

For $0\leq \nu\leq 1$, the $\nu$-weighted geometric mean, denoted by ${\bf A}\sharp_{\nu} {\bf B}$, is defined as

The $\nu$-weighted geometric mean was introduced by Kubo and Ando ^[11], and when $\nu = \frac{1}{2}$ this is just the geometric mean. One can show that ${\bf A}\sharp_{\nu} {\bf B} = {\bf B}\sharp_{1-\nu} {\bf A}$ for $0\leq \nu\leq 1$. When ${\bf A}$ and ${\bf B}$ commute, ${\bf A}\sharp_{\nu} {\bf B} = {\bf A}^{\nu} {\bf B}^{1-\nu}$. The $\nu$-weighted arithmetic mean of ${\bf A}$ and ${\bf B}$, denoted by ${\bf A}\nabla_{\nu} {\bf B}$, is defined as

Theorem 2.16. Let ${\bf A}, {\bf B}, {\bf C}, {\bf D}\in \mathcal{B}(\mathcal{H})$, and let $\psi$ be a non-negative increasing convex function on $[0, \infty)$. If for given $m', M' > 0$,

then

where

To prove Theorem 2.16, we need the following result that established by Furuichi ^[9].

Corollary 2.17. Let $0 \leq \nu \leq 1$, $-1 \leq r_1 < 0$, $0 < r_2 \leq 1$ and let ${\bf T}$ and ${\bf S}$ be strictly positive operators satisfying (i) $0 < m \leq {\bf T} \leq m' < M' \leq {\bf S} \leq M$ or (ii) $0 < m \leq {\bf S} \leq m' < M' \leq {\bf T} \leq M$ with $h = \frac{M}{m}$ and $h' = \frac{M'}{m'}$. Then

Proof of Theorem 2.16. From Corollary 2.17, we have

for ${\bf T}, {\bf S} > 0$ with $m', M' > 0$ satisfying $0 < m' \leq {\bf T} \leq {\bf S} \leq M'$ or $0 < m' \leq {\bf S} \leq {\bf T} \leq M'$, where $exp_r (\xi) : = (1 + r\xi)^{1/r}$, if $1 + r\xi > 0$, and it is undefined otherwise. Since $exp_r(\xi)$ is decreasing in $r\in [-1, 0)$, the above inequality gives a tight lower bound when $r = -1$. After all, we have the scalar inequality:

for $\sigma, \tau > 0$ and $m', M' > 0$ such that $0 < m' \leq \min\{\sigma, \tau \} \leq \max\{\sigma, \tau \} \leq M'$. Applying this inequality with a similar argument as in Theorem 2.12, we obtain the desired result. □

Theorem 2.18. Let ${\bf A}, {\bf B}, {\bf C}, {\bf D}\in \mathcal{B}(\mathcal{H})$ $0 < \nu < 1$ and let $\psi$ be a non-negative increasing convex function on $[0, \infty)$. Then

where $r = \min\{\nu, 1-\nu\}$ and

Proof. We assume $0\leq \nu\leq \frac{1}{2}$. For each unit vector $\xi\in \mathcal{H}$,

Hence

If we apply similar arguments for $\frac{1}{2}\leq \nu\leq 1$, then we can write

We know that if ${\bf T}\in \mathcal{B}(\mathcal{H})$ is a positive operator, then $\left\Vert{ {\bf T}}\right\Vert = \sup_{\left\Vert{\xi}\right\Vert = 1}\left < { {\bf T}\xi, \xi}\right >$. By using this, the continuity and the increase of $\psi$, we have

On the other hand, if ${\bf X}\in \mathcal{B}(\mathcal{H})$, and if $\psi$ is a non-negative increasing function on $[0, \infty)$, then $\psi(\left\Vert{ {\bf X}}\right\Vert) = \left\Vert{\psi(| {\bf X}|)}\right\Vert$.

Now from the proof of Theorem 2.9, we have

This completes the proof. □

Inequality (2.18) includes several numerical radius inequalities as special cases.

Corollary 2.19. Let ${\bf T}\in \mathcal{B}(\mathcal{H})$, $\alpha+\beta\geq 1$, $0 < \nu < 1$ and let $\psi$ be a non-negative increasing convex function on $[0, \infty)$. Then

where $r$ and $\gamma(\psi)$ as in Theorem 2.18.

Proof. Let ${\bf T} = U| {\bf T}|$ be the polar decomposition of the operator ${\bf T}$, where $U$ is partial isometry and the kernel $\ker(U) = N(| {\bf T}|)$. If we take ${\bf D} = U, {\bf C} = | {\bf T}|^{\beta}, {\bf B} = U$ and ${\bf A} = | {\bf T}|^{\alpha}$, we have

So, the result follows by Theorem 2.18. □

Corollary 2.20. Let ${\bf T}\in \mathcal{B}(\mathcal{H})$, $\alpha, \beta\geq 0$ such that $\alpha+\beta\geq 2$, $0 < \nu < 1$ and let $\psi$ be a non-negative increasing convex function on $[0, \infty)$. Then

where $r$ and $\gamma(\psi)$ as in Theorem 2.18.

Proof. Let ${\bf T}^* = U| {\bf T}^*|$ be the polar decomposition of the operator ${\bf T}^*$, where $U$ is partial isometry and the kernel $\ker(U) = N(| {\bf T}|)$. Then ${\bf T} = | {\bf T}^*|U^*$. If we take ${\bf D} = U, {\bf C} = | {\bf T}^*|^{\beta}, {\bf B} = | {\bf T}^*|^{\alpha}$ and ${\bf A} = U^*$, we have

So, the result follows by Theorem 2.18. □

Corollary 2.21. Let ${\bf T}\in \mathcal{B}(\mathcal{H})$, $\alpha, \beta\geq 0$, $0 < \nu < 1$ and let $\psi$ be a non-negative increasing convex function on $[0, \infty)$. Then

where $r$ and $\gamma(\psi)$ as in Theorem 2.18.

Proof. In Theorem 2.18, if we let ${\bf D} = | {\bf T}|^{\alpha}, {\bf C} = {\bf T}, {\bf B} = {\bf T}$ and ${\bf A} = | {\bf T}|^{\beta}$, then

so the result: □

Inequalities for numerical radius and operator norm have now been given, although in the context of superquadratic functions. Remember that a function $\psi:[0, \infty) \longrightarrow \mathbb R$ is termed superquadratic if there exists a constant $C_x\in \mathbb R$ such that

for all $t\geq 0$. If $-\psi$ is superquadratic, we say $\psi$ is subquadratic. As a result, for a superquadratic function, $\psi$ must be above its tangent line plus a translation of $\psi$. Superquadratic functions appear to be stronger than convex functions at first glance, however they may be deemed weaker if $\psi$ has negative values. If $\psi$ is superquadratic and non-negative, then $\psi$ is increasing and convex, and if $C_{\xi}$ is equal to (2.19), then $C_{\xi}\geq 0$ ^[1].

Theorem 2.22. Let ${\bf A}\in \mathcal{B}(\mathcal{H})$ and let $\psi$ be a non-negative superquadratic function. Then

Proof. Letting $\xi = \left\Vert{ {\bf A}}\right\Vert$ in the inequality (2.19), we get

By applying functional calculus for the operator $| {\bf A}|$ in (2.21) we get

Hence,

Consequently,

for every unit vector $\xi\in \mathcal{H}$.

Now, by taking supremum over $\xi\in \mathcal{H}$ with $\left\Vert{\xi}\right\Vert = 1$ in (2.23), and using the fact $w(| {\bf A}|) = \left\Vert{ {\bf A}}\right\Vert\geq w({\bf A})$, and $\psi$ is increasing, we deduce the desired inequality (2.20). □

Applying Theorem 2.22 to the superquadratic function $\psi(t) = t^r$ $(r \geq 2)$, we reach the following corollary:

Corollary 2.23. Let $A\in \mathcal{B}(\mathcal{H})$. Then for any $r\geq 2$,

In particular

3.   Further refinements of numerical radius inequalities

In this section, We provide various inequalities involving power numerical radii and the usual operator norms of Hilbert space operators. In particular, if ${\bf A}_i, {\bf B}_i$ and ${\bf X}_i$ are bounded linear operators ($i = 1, 2, \cdots n\in \mathbb N$), then we estimate the numerical radius to $\sum_{i = 1}^{m} {\bf X}_i {\bf A}_i^m {\bf B}_i$ for some $m\in \mathbb N$.

The following lemma is a straightforward application of Jensen's inequality about the convexity or concavity of certain power functions. Schlömilch's inequality for the weighted means of non-negative real numbers is a specific example of this inequality.

Lemma 3.1. Let $\sigma, \tau > 0$ and $0\leq \alpha\leq 1$. Then

The following result was established by Kittaneh and Manasrah ^[16], which is a refinement of the scalar Young inequality.

Lemma 3.2. Let $\sigma, \tau > 0$, and $p, q > 1$ such that $\frac{1}{p}+\frac{1}{q} = 1$. Then

where $r_0 = \min\{\frac{1}{p}, \frac{1}{q}\}$.

Manasrah and Kittaneh have generalized (3.2) in ^[17], as follows:

Lemma 3.3. Let $\sigma, \tau > 0$, and $p, q > 1$ such that $\frac{1}{p}+\frac{1}{q} = 1$. Then for $m = 1, 2, \cdots,$ we have

where $r_0 = \min\{\frac{1}{p}, \frac{1}{q}\}$. In particular, if $p = q = 2$, then

For $m = 1$, and $p = q = 2$, we have

The convexity of the function $\psi(t) = t^r$, $r\geq 1$ leads to the following lemma, which deals with positive real numbers.

Lemma 3.4. Let $\sigma_i, i = 1, \cdots, n$ be positive real numbers. Then

Theorem 3.5. Let ${\bf A}_i, {\bf C}_i, D_i\in \mathcal{B}(\mathcal{H})$, $(i = 1, \cdots, n)$, $m\in \mathbb N$. Then

for all $r\geq 1$.

Proof. Let $\xi\in \mathcal{H}$ be any unit vector. Then by Lemma 2.5, Lemma 3.1 and Lemma 3.4, we obtain that

This implies that

Taking the supremum over all unit vectors $\xi\in \mathcal{H}$, we get the result. □

For ${\bf D}_i = {\bf A}_i = I$ in inequality (3.7), we have:

Corollary 3.6. Let ${\bf C}_i\in \mathcal{B}(\mathcal{H})$, ($i = 1, \cdots, n$), $m\in \mathbb N$. Then

for all $r\geq 1$.

The following is an example of how Corollary 3.6 may be used. It entails a numerical radius inequality for operator powers.

Corollary 3.7. Let ${\bf C}\in \mathcal{B}(\mathcal{H})$ and $m\in \mathbb N$. Then for all $r\geq 1$, we have

Theorem 3.8. Let ${\bf A}_i, {\bf C}_i, {\bf D}_i\in \mathcal{B}(\mathcal{H})$, $(i = 1, \cdots, n)$, $m\in \mathbb N$ and $0\leq \alpha\leq 1$. Then

for all $r\geq 1$.

Proof. Let $\xi\in \mathcal{H}$ be any unit vector. Then by Lemmas 2.5, 3.1 and 3.4, we obtain

Hence,

Taking the supremum over all unit vectors $\xi\in \mathcal{H}$, we deduce the desired result. □

Theorem 3.9. Let ${\bf A}_i, {\bf C}_i, {\bf D}_i\in \mathcal{B}(\mathcal{H})$, $(i = 1, \cdots, n)$, $m\in \mathbb N$ and $p, q > 1$ such that $\frac{1}{p}+\frac{1}{q} = 1$. Then

where $r_0 = \min\{\frac{1}{p}, \frac{1}{q}\}$ and

Proof. Let $\xi\in \mathcal{H}$ be any unit vector. Then by Lemmas 2.5, 3.2 and 3.4, we obtain

This implies that

Taking the supremum over all unit vectors $\xi\in \mathcal{H}$, we deduce the desired result. □

Theorem 3.10. Let ${\bf A}_i, {\bf C}_i, {\bf D}_i\in \mathcal{B}(\mathcal{H})$, $(i = 1, \cdots, n)$, $m\in \mathbb N$ and $p, q > 1$ such that $\frac{1}{p}+\frac{1}{q} = 1$ and $k = 1, 2, \cdots$. Then

where $r_0 = \min\{\frac{1}{p}, \frac{1}{q}\}$ and

for all $r\geq 1$.

Proof. Let $\xi\in \mathcal{H}$ be any unit vector. Then by Lemmas 2.5, 3.3 and 3.4, we obtain

This implies that

Taking the supremum over all unit vectors $\xi\in \mathcal{H}$, we deduce the desired result. □

For $k = 1$, and $p = q = 2$, we have:

Corollary 3.11. Let ${\bf A}_i, {\bf C}_i, {\bf D}_i\in \mathcal{B}(\mathcal{H})$, $(i = 1, \cdots, n)$, $m\in \mathbb N$. Then

where

for all $r\geq 1$.

The following lemma is an extended variant of the mixed Schwarz inequality, which has been shown by Kittaneh ^[12] and is highly relevant in the following results.

Lemma 3.12. Let ${\bf A}\in \mathcal{B}(\mathcal{H})$, and $\psi$ and $\phi$ be non-negative functions on $[0, \infty)$ which are continuous such that $\psi(t)\phi(t) = t$ for all $t\in [0, \infty)$. Then

for all $\xi, \zeta\in \mathcal{H}$.

The next results give improvements of the inequality (1.10).

Theorem 3.13. Let ${\bf A}_i, {\bf B}_i, {\bf X}_i\in \mathcal{B}(\mathcal{H})$, $(i = 1, \cdots, n)$, $m\in \mathbb N$, $p, q > 1$ with $\frac{1}{p}+\frac{1}{q} = 1$ and let $\psi$ and $\phi$ be as in Lemma 3.12. Then for all $r\geq 1$, we have

where $r_0 = \min\{\frac{1}{p}, \frac{1}{q}\}$, $S_{i, j} = {\bf X}_i\psi^2\left({\left\vert{ {\bf A}_i^{j*}}\right\vert}\right) {\bf X}_i^*$, $T_{i, j} = \left({ {\bf A}_i^{m-j} {\bf B}_i}\right)^*\phi^2\left({\left\vert{ {\bf A}_i^j}\right\vert}\right) {\bf A}_i^{m-j} {\bf B}_i$ and

Proof. Let $\xi\in \mathcal{H}$ be any unit vector. Then by Lemma 3.3, Lemma 3.4 and Lemma 3.12, we obtain

and so

Hence,

Taking the supremum over all unit vectors $\xi\in \mathcal{H}$, we deduce the desired result. □

Inequality (3.17) includes several numerical radius inequalities as special cases. Samples of inequalities are demonstrated in what follows, for $\psi(t) = t^{\lambda}$ and $\phi(t) = t^{1-\lambda}$, $\lambda\in (0, 1)$ in inequality (3.17).

Corollary 3.14. Let ${\bf A}_i, {\bf B}_i, {\bf X}_i\in \mathcal{B}(\mathcal{H})$, $(i = 1, \cdots, n)$, $m\in \mathbb N$, $p, q > 1$ with $\frac{1}{p}+\frac{1}{q} = 1$ and let $\psi$ and $g$ be as in Lemma 3.12. Then for all $r\geq 1$, we have

where $r_0 = \min\{\frac{1}{p}, \frac{1}{q}\}$, $S_{i, j} = {\bf X}_i\left\vert{ {\bf A}_i^{j*}}\right\vert^{2\lambda} {\bf X}_i^*$, $T_{i, j} = \left({ {\bf A}_i^{m-j} {\bf B}_i}\right)^*\left\vert{ {\bf A}_i^j}\right\vert^{2(1-\lambda)} {\bf A}_i^{m-j} {\bf B}_i$ and

For ${\bf X}_i = {\bf B}_i = I$ in inequality (3.14) we get the following numerical radius inequality.

Corollary 3.15. Let ${\bf A}_i, \in \mathcal{B}(\mathcal{H})$, $(i = 1, \cdots, n)$, $m\in \mathbb N$, $p, q > 1$ with $\frac{1}{p}+\frac{1}{q} = 1$ and let $\psi$ and $g$ be as in Lemma 3.12. Then for all $r\geq 1$, we have

where $r_0 = \min\{\frac{1}{p}, \frac{1}{q}\}$, $S_{i, j} = \psi^2\left({\left\vert{\left({ {\bf A}_i^{j}}\right)^*}\right\vert}\right)$, $T_{i, j} = \left({ {\bf A}_i^{m-j}}\right)^{*}\phi^2\left({\left\vert{ {\bf A}_i^j}\right\vert}\right) {\bf A}_i^{m-j}$ and

An application of Corollary 3.15 can be seen in the following result. It involves a numerical radius inequality for the powers of operator.

Corollary 3.16. Let ${\bf A}\in \mathcal{B}(\mathcal{H})$, $m\in \mathbb N$, $p, q > 1$ with $\frac{1}{p}+\frac{1}{q} = 1$ and let $\psi(t) = t^{\lambda}$ and $\phi(t) = t^{1-\lambda}$. Then for all $r\geq 1$, we have

where $r_0 = \min\{\frac{1}{p}, \frac{1}{q}\}$, $S_{j} = \left\vert{\left({A^{j}}\right)^*}\right\vert^{2\lambda}$, $T_{j} = \left({ {\bf A}^{m-j}}\right)^{*}\left\vert{ {\bf A}^j}\right\vert^{2(1-\lambda)} {\bf A}^{m-j}$ and

Theorem 3.17. Let ${\bf A}_i, {\bf B}_i, {\bf X}_i\in \mathcal{B}(\mathcal{H})$, $(i = 1, \cdots, n)$, $m, k\in \mathbb N$, $p, q > 1$ with $\frac{1}{p}+\frac{1}{q} = 1$ and let $\psi$ and $\phi$ be as in Lemma 3.12. Then for all $r\geq 1$, we have

where $r_0 = \min\{\frac{1}{p}, \frac{1}{q}\}$, $S_{i, j} = {\bf X}_i\psi^2\left({\left\vert{ {\bf A}_i^{j*}}\right\vert}\right) {\bf X}_i^*$, $T_{i, j} = \left({ {\bf A}_i^{m-j} {\bf B}_i}\right)^*\phi^2\left({\left\vert{ {\bf A}_i^j}\right\vert}\right) {\bf A}_i^{m-j} {\bf B}_i$ and

Proof. Let $\xi\in \mathcal{H}$ be any unit vector. Then by Lemmas 3.3, 3.4 and 3.12, we obtain

this implies that

Hence,

Taking the supremum over all unit vectors $\xi\in \mathcal{H}$, we deduce the desired result. □

If we take $k = 1$ and $p = q$, we have:

Corollary 3.18. Let ${\bf A}_i, {\bf B}_i, {\bf X}_i\in \mathcal{B}(\mathcal{H})$, $(i = 1, \cdots, n)$, $m\in \mathbb N$, and let $\psi$ and $\phi$ be as in Lemma 3.12. Then for all $r\geq 1$, we have

where $S_{i, j} = {\bf X}_i\psi^2\left({\left\vert{ {\bf A}_i^{j*}}\right\vert}\right) {\bf X}_i^*$, $T_{i, j} = \left({ {\bf A}_i^{m-j} {\bf B}_i}\right)^*\phi^2\left({\left\vert{ {\bf A}_i^j}\right\vert}\right) {\bf A}_i^{m-j} {\bf B}_i$ and

4.   Conclusions

In this work, we have derived a series of precise inequalities involving the standard operator norms of Hilbert space operators and powers of the numerical radii. These inequalities build upon traditional convexity inequalities for nonnegative real numbers and extend earlier numerical radius inequalities in the context of operator theory.

As part of future work, further investigations could explore the practical implications and applications of these inequalities in the context of operator theory and related areas of mathematics and physics.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We thank the referees for their valuable comments and helpful suggestions.

Conflict of interest

The authors declare no conflict of interests.