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

We provide a number of sharp inequalities involving the usual operator norms of Hilbert space operators and powers of the numerical radii. Based on the traditional convexity inequalities for nonnegative real numbers and some generalize earlier numerical radius inequalities, operator. Precisely, we prove that if $\A_i,\B_i,\X_i\in\bh$ ($i=1,2,\cdots,n$), $m\in\N$, $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$ 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 \begin{equation*} w^{2r}\bra{\sum_{i=1}^{n}\X_i\A_i^m\B_i}\leq \frac{n^{2r-1}}{m}\sum_{j=1}^{m}\norm{\sum_{i=1}^{n}\frac{1}{p}S_{i,j}^{pr}+\frac{1}{q}T_{i,j}^{qr}}-r_0\inf_{\norm{x}=1}\rho(\xi), \end{equation*} where $r_0=\min\{\frac{1}{p},\frac{1}{q}\}$, $S_{i,j}=\X_i\phi^2\bra{\abs{\A_i^{j*}}}\X_i^*$, $T_{i,j}=\bra{\A_i^{m-j}\B_i}^*\psi^2\bra{\abs{\A_i^j}}\A_i^{m-j}\B_i$ and $$\rho(x)=\frac{n^{2r-1}}{m}\sum_{j=1}^{m}\sum_{i=1}^{n}\bra{\seq{S_{i,j}^r\xi,\xi}^{\frac{p}{2}}-\seq{T_{i,j}^r\xi,\xi}^{\frac{q}{2}}}^2.$$

1 p + 1 q = 1 and φ and ψ are non-negative functions on [0, ∞) which are continuous such that φ(t)ψ(t) = t for all t ∈ [0, ∞), then where r 0 = min{ 1 p , 1 q }, S i,j = X i φ 2 A j * i 1 Introduction Let H be complex Hilbert space and B(H) be the C * -algebra of all bounded linear operator on H.An operator T ∈ B(H) is said to be positive if Tξ, ξ ≥ 0 holds for all ξ ∈ H.We write T ≥ 0 if T is positive.
The numerical radius of T ∈ B(H) is defined by It is well known that w(•) defines a norm on B(H), which is equivalent to the usual operator norm • .In fact, for any T ∈ B(H), 1 2 T ≤ w(T) ≤ T .
(1) Also, if T ∈ B(H) is normal, then w(T) = T .An important inequality for w(T) is the power inequality stating that w(T n ) ≤ (w(T)) n for every natural numbers n.
A general numerical radius inequality has been established by Kittaneh [2005], it has been proved that if A, B, C, D, T, S ∈ B(H), then for all α ∈ (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, Gustafson and Rao [1997] and Rashid [2019], Rashid andAltaweel [2021, 2022], Rashid [2023]).If A, B ∈ B(H), then w(AB) ≤ 4w(A)w(B).
In the case that N M = M N , we have w(AB) ≤ 2w(A)w(B).
Moreover, if N and M are normal, then w(AB) ≤ w(A)w(B).
Recently, Dragomir [2009] proved that if N, M ∈ B(H) and r ≥ 1, then Shebrawi and Albadawi [2009] discovered a fascinating numerical radius inequality, it has been shown that if A, X, B ∈ B(H), then Very recently, Aldolat and Al-Zoubi [2016], showed that if 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 and X i are bounded linear operators, we estimate the numerical radius to m j=1

Inequalities for sums and products of operators
In this part, we built a generic numerical radius inequality for Hilbert space operators that results in well-known new numerical radius inequalities as an example.A norm inequality operator sets off this section.In fact, we give B * A + DC * an additional upper bound.However, the following lemma is crucial to the theorem's proof.Lemma 2.1 (Dragomir [2006b]).Let ξ, ζ, η ∈ H. Then we have Proof.For ξ, ζ ∈ H, we have by triangle inequality, we have we have Combining the inequalities ( 12) and ( 13), we have Taking the supremum over all unit vectors ξ, ζ, we obtain the desired inequality.
In Theorem 2.2, if we let A = B = C = D = S, we have Corollary 2.3.Let S ∈ B(H).Then In the proof of Theorem 2.2, if we let ξ = ζ, we have Corollary 2.4.Let A, B, C, D ∈ B(H).Then The following lemma gives a basic but useful extension for four operators of the Schwarz inequality due to Dragomir Dragomir [2014].Lemma 2.5.Let A, B, C, D ∈ B(H).Then for ξ, ζ ∈ H we have the inequality The equality case holds if and only if the vectors BAξ and C * D * ζ are linearly dependent in 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 Kittaneh [1988]).Lemma 2.6.Let T ∈ B(H), T ≥ 0 and let ξ ∈ H be any unit vector.Then we have (i) Tξ, ξ r ≤ T r ξ, ξ for r ≥ 1.
The next result is well known in the literature as the Mond-Pečarić inequality Hosseini et al. [2019].Lemma 2.7.If ψ is a convex function on a real interval J containing the spectrum of the self-adjoint operator T, then for any unit vector ξ ∈ H, ψ( Tξ, ξ ) ≤ ψ(T)ξ, ξ (14) and the reverse inequality holds if ψ is concave.
The forth lemma is a direct consequence of [Aujla and Silva, 2003, Theorem 2.3].Lemma 2.8.Let ψ be a non-negative non-decreasing convex function on [0, ∞) and let T, S ∈ H be positive operators.Then for any 0 < µ < 1, The above four lemmas admit the following more general result.Theorem 2.9.Let A, B, C, D ∈ B(H).If ψ is a non-negative increasing convex function on [0, ∞), then for any In particular, for all r ≥ 1.
Taking the supremum over ξ ∈ H with ξ = 1, we infer that On account of assumptions on ψ, we can write The inequality (17) follows directly from ( 16) by taking ψ(t) = t r (r ≥ 1).
(i) If we let ψ(t) = t r (r ≥ 1), we have (ii) Letting D = S * , A = T and let ψ(t) = t r (r ≥ 1), we have We give an example to clarify part (ii) in Remark 2.14 Example 2.15.
= 16.0625.If we take δ = 0.3 and ∆ = .4,then Recall that the weighted operator arithmetic mean ∇ ν and geometric mean ν , for 0 < ν < 1, positive invertible operator A, and positive operator B, are defined as follows: we denote the arithmetic and geometric means, respectively, by ∇ and .
Theorem 2.16.Let A, B, C, D ∈ B(H), and let f be a non-negative increasing convex function on [0, ∞). where To prove Theorem 2.16, we need the following result that established by Furuichi [2019].
If we apply similar arguments for 1 2 ≤ ν ≤ 1, then we can write We know that if T ∈ B(H) is a positive operator, then T = sup ξ =1 Tξ, ξ .By using this, the continuity and the increase of ψ, we have On the other hand, if X ∈ B(H), and if ψ is a non-negative increasing function on [0, ∞), then ψ( X ) = ψ(|X|) .Now from the proof of Theorem 2.9, we have This completes the proof.Inequality (28) includes several numerical radius inequalities as special cases.Corollary 2.19.Let T ∈ B(H), α + β ≥ 1, 0 < ν < 1 and let ψ be a non-negative increasing convex function on [0, ∞).Then where r and γ(ψ) as in Theorem 2.18.
Proof.Let T = U |T| be the polar decomposition of the operator T, where U is partial isometry and the kernel ker(U ) = N (|T|).If we take D = U, C = |T| β , B = U and A = |T| α , we have So, the result follows by Theorem 2.18.
Proof.In Theorem 2.18, if we let D = |T| α , C = T, B = T and A = |T| β , 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 for all t ≥ 0. If −ψ is superquadratic, we say ψ is subquadratic.As a result, for a superquadratic function, ψ must be above its tangent line plus a translation of ψ.Superquadratic functions appear to be stronger than convex functions at first glance, however they may be deemed weaker if ψ has negative values.If ψ is superquadratic and non-negative, then Then ψ is increasing and convex, and if C ξ is equal to (29), then C ξ ≥ 0 Abramovich et al. [2004].
Theorem 2.22.Let A ∈ B(H) and let ψ be a non-negative superquadratic function.Then Proof.Letting ξ = A in the inequality (29), we get By applying functional calculus for the operator |A| in (31) we get Consequently, for every unit vector ξ ∈ H. Now, by taking supremum over ξ ∈ H with ξ = 1 in (33), and using the fact w(|A|) = A ≥ w(A), and ψ is increasing, we deduce the desired inequality (30).

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 A i , B i and X i are bounded linear operators (i = 1, 2, • • • n ∈ N) , then we estimate the numerical radius to m j=1 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 σ, τ > 0 and 0 ≤ α ≤ 1.Then The following result was established by Kittaneh and Manasrah [2010], which is a refinement of the scalar Young inequality.
Proof.Let ξ ∈ H be any unit vector.Then by Lemma 2.5, Lemma 3.1 and Lemma 3.4, we obtain that Taking the supremum over all unit vectors ξ ∈ H, we get the result.
for all r ≥ 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 C ∈ B(H) and m ∈ N. Then for all r ≥ 1, we have for all r ≥ 1.
Proof.Let ξ ∈ H be any unit vector.Then by Lemma 2.5, Lemma 3.1 and Lemma 3.4, we obtain Taking the supremum over all unit vectors ξ ∈ H, we deduce the desired result.
Theorem 3.9. where Proof.Let ξ ∈ H be any unit vector.Then by Lemma 2.5, Lemma 3.2 and Lemma 3.4, we obtain This implies that Taking the supremum over all unit vectors ξ ∈ H, we deduce the desired result.
Proof.Let ξ ∈ H be any unit vector.Then by Lemma 2.5, Lemma 3.3 and Lemma 3.4, we obtain Taking the supremum over all unit vectors ξ ∈ H, we deduce the desired result.
For k = 1, and p = q = 2, we have where 2 for all r ≥ 1.
The following lemma is an extended variant of the mixed Schwarz inequality, which has been shown by Kittaneh [1988] and is highly relevant in the following results.
The next results give improvements of the inequality (10).
and let ψ and φ be as in Lemma 3.12.Then for all r ≥ 1, we have where Proof.Let ξ ∈ H be any unit vector.Then by Lemma 3.3, Lemma 3.4 and Lemma 3.12, we obtain Taking the supremum over all unit vectors ξ ∈ H, we deduce the desired result.
and let ψ and g be as in Lemma 3.12.Then for all r ≥ 1, we have where For X i = B i = I in inequality (50) we get the following numerical radius inequality Corollary 3.15.
and let ψ and g be as in Lemma 3.12.Then for all r ≥ 1, we have where 2 .
An application of Corollary 3.15 can be seen in the following result.It involves a numerical radius inequality for the powers of operator.
If we take k = 1 and p = q, we have Corollary 3.18.Let A i , B i , X i ∈ B(H), (i = 1, • • • , n), m ∈ N, and let ψ and φ be as in Lemma 3.12.Then for all r ≥ 1, we have

)
where r and γ(ψ) as in Theorem 2.18.Proof.Let T * = U |T * | be the polar decomposition of the operator T * , where U is partial isometry and the kernel ker(U ) = N (|T|).Then T = |T * |U * .If we take D = U, C = |T * | β , B = |T * | α and A = U * , we have Let ξ ∈ H be any unit vector.Then by Lemma 3.3, Lemma 3.4 and Lemma 3.12, we obtain