# On Numerical Radius Bounds Involving Generalized Aluthge Transform

Authors: Tao Yan; Javariya Hyder; Muhammad Saeed Akram; Ghulam Farid; Kamsing Nonlaopon
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we establish some upper bounds of the numerical radius of a bounded linear operatorS defined on a complex Hilbert space with polar decomposition S=U∣S∣, involving generalized Aluthge transform. These bounds generalize some bounds of the numerical radius existing in the literature. Moreover, we consider particular cases of generalized Aluthge transform and give some examples where some upper bounds of numerical radius are computed and analyzed for certain operators.


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## Body

## 1. Introduction

In mathematical analysis, inequalities play a vital role in studying the properties of operators in the form of their upper and lower bounds. Mathematical inequalities provide the best way to describe as well as propose solutions to real-world problems in almost all fields of science and engineering. The boundedness property of different kinds of operators studied in the subjects of analysis, precisely in mathematical and functional analysis, is the key factor in developing the theory and applications. For example, upper and lower bounds are utilized to define the operator norm, which plays significantly in solving related problems. The study of the numerical radius of an operator defined on the Hilbert space is in the focus of researchers in these days in studying perturbation, convergence, iterative solution methods, and integrative methods, etc, see [1–9]. In this regard, the numerical radius inequality stated in (3) is studied extensively by various mathematicians, see [10–21]. Actually, it is interesting for the researchers to get refinements and generalizations of this inequality [22–27]. The goal of this paper is to study generalizations of numerical radius bounds under certain additional conditions. Henceforth, we define the preliminary notions to proceed with the findings of this work.The polar decomposition is an important feature in the theory of operators. It is defined byA=UB, where U is the unitary matrix, and B is the symmetric positive semidefinite matrix. It is interesting to see that when A is nonsingular and symmetric, then B is a good symmetric positive definite approximation to A and 1/2A+B is the best symmetric positive semidefinite approximation to A, see [6]. Let BH be the C∗-algebra of all bounded linear operators on complex Hilbert space. Let S=U∣S∣ be the unique polar decomposition of S∈BH, where U is a partial isometry and ∣S∣ is the square root of an operator which is defined as S=S∗S. The numerical range of an operator S is defined as
(1)WS=Sx,x:x=1,x∈H,where WS denotes the numerical range. The numerical radius of an operator is the radius of the smallest circle centered at the origin and contains the numerical range, i.e.,
(2)wS=supλ:λ∈WS.The numerical radius defines a norm onBH which is equivalent to the usual operator norm, satisfying the following inequality:
(3)12S≤wS≤S.IfS2=0, then the first inequality becomes equality and if S is normal then the second inequality becomes equality. Many authors worked on numerical radius inequalities and developed a number of numerical radius bounds [10, 13–18].In [17], Kittaneh gave an upper bound of numerical radius as follows:
(4)wS≤12S+S21/2,and showed that this bound is sharper than the upper bound given in (3).In [25], Aluthge introduced a transform of an operator S∈BH which is called Aluthge transform that is defined as
(5)ΔS=S1/2US1/2.In [26], Yamazaki developed an upper bound of the numerical radius involving Aluthge transform as follows:
(6)wS≤12S+wΔS,and proved that it is sharper than the bound given in (4).In [27], Okubo introduced a new generalization of Aluthge transform, called λ-Aluthge transform defined by
(7)ΔλS=SλUS1−λ;λ∈0,1.In [23], Abu-Omar and Kittaneh further generalized the bound given in (6) using λ-Aluthge transform as follows:
(8)wS≤12S+wΔλS.In [19], Bhunia et al. found some bounds of the numerical radius for S∈BH. Later, Bag et al. [24] working along the same lines succeeded to get the following upper bounds of the numerical radius:
(9)w2S≤12SΔλS+14S∗S+SS∗,(10)w2S≤14wΔλS2+SΔλS+S∗S+SS∗,(11)w2S≤∑n=1∞14nΔλn−1SΔλnS+Δλn−1S∗Δλn−1S+Δλn−1SΔλn−1S∗,(12)w4S≤116wΔλS2+SΔλS2+18wS2P+PS2+116P2,where P=S∗S+SS∗ and λ∈0,1.In [22], Shebrawi and Bakherad presented a new form of Aluthge transform so called generalized Aluthge transform defined by
(13)Δf,gS=fSUgS,where f and g are nonnegative continuous functions such that fSgS=S,S≥0. They proved the following upper bound of the numerical radius by using generalized Aluthge transform
(14)wS≤12S+wΔf,gS,which is a generalization of the upper bound shown in (6) and (8).Our aim is to study the upper bounds of the numerical radius by applying generalized Aluthge transform defined in (13) by imposing further certain conditions on continuous functions. The first contribution of this paper is that we develop upper bounds of the numerical radius using generalized Aluthge transform, which extends and generalizes some already existing bounds. Specifically, we extend the inequalities (9)–(12) for generalized Aluthge transform under certain conditions on f and g. As a consequence, the upper bounds of numerical radius involving Aluthge transform and λ-Aluthge transform appear as a special case of our bounds. Another contribution of the paper is that we have presented examples of generalized Aluthge transform in addition to the classical Aluthge transform and λ-Aluthge transform, which are used for computing bounds of numerical radius. More precisely, we have considered five choices of continuous functions f and g in (13) and used them to compute upper bounds of numerical radius for certain operators.


## 2. Main Results

We start this section by attaining the generalized Aluthge transformΔf,g defined in (13) under the following additional conditions:
(i)
gSfS=S(ii)
fS and gS both are positive operatorsNow, we give some results that will be used repeatedly to achieve our goal.Lemma 1 [26].
LetS∈BH. Then, we have
(15)wS=supθ∈ℝHθ=supθ∈ℝReeιθS,where Hθ=ReeιθS=eιθS+e−ιθS∗/2 for all θ∈ℝ.Lemma 2 [23].
LetM1,M2,N1,N2∈BH. Then,
(16)rM1N1+M2N2≤12wN1M1+wN2M2+12wN1M1−wN2M2+4N1M2N2M1,where r denotes the spectral radius.Next, we give the numerical radius bound by using the generalized Aluthge transform.Theorem 3.
LetS∈BH. Then, we have
(17)w2S≤12gSΔf,gSfS+14S∗S+SS∗.Proof.
Since(18)Hθ=12eιθS+e−ιθS∗forallθ∈ℝ,therefore,
(19)Hθ2=14eιθS+e−ιθS∗2=14e2ιθS2+e−2ιθS∗2+SS∗+S∗S=14e2ιθUSUS+e−2ιθSU∗SU∗+SS∗+S∗S=14e2ιθUgSfSUgSfS+e−2ιθfSgSU∗fSgSU∗+SS∗+S∗S=14e2ιθUgSΔf,gSfS+e−2ιθfSΔf,gS∗gSU∗+SS∗+S∗S.
The third equality is obtained by puttingS=U∣S∣ and S∗=∣S∣U∗ in second equality, the fourth equality holds because fSgS=S and g∣S∣f∣S∣=∣S∣, and the fifth equality holds because Δf,gS=f∣S∣Ug∣S∣ and Δf,gS∗=g∣S∣U∗f∣S∣. Since ZZ∗=Z2 for any Z∈BH, therefore
(20)Hθ2=14e2ιθUgSΔf,gSfS+e−2ιθfSΔf,gS∗gSU∗+SS∗+S∗S≤14gSΔf,gSfS+fSΔf,gS∗gS+SS∗+S∗S=142gSΔf,gSfS+SS∗+S∗S.
The first inequality holds becauseS1S2≤S1S2, S1+S2≤S1+S2 for any S1,S2∈BH, U is partial isometry and ∣e2ιθ∣=1 and the second equality holds by using the fact that S=S∗.
Now, by taking supremum of the last inequality overθ∈ℝ and then using Lemma 1, we get
(21)w2S≤12gSΔf,gSfS+14SS∗+S∗S,as required.The following result is another generalized bound of numerical radius for bounded linear operators onH.Theorem 4.
LetS∈BH. Then,
(22)w2S≤14wΔf,gS2+gSΔf,gSfS+S∗S+SS∗.Proof.
LetS be any bounded linear operator with polar decomposition S=US. Since
(23)Hθ=12eιθS+e−ιθS∗forallθ∈ℝ,therefore,
(24)Hθ2=14eιθS+e−ιθS∗2=14e2ιθS2+e−2ιθS∗2+SS∗+S∗S.
Using the properties of operator norm· on BH, we have
(25)Hθ2≤14e2ιθUgSΔf,gSfS+e−2ιθfSΔf,gS∗gSU∗+SS∗+S∗S=14re2ιθUgSΔf,gSfS+e−2ιθfSΔf,gS∗gSU∗+SS∗+S∗S=14rM1N1+M2N2+SS∗+S∗S,where M1=eιθUg∣S∣Δf,gS, N1=f∣S∣, M2=e−2ιθf∣S∣Δf,gS∗,N2=g∣S∣U∗. The first equality above holds for hermitian operator A∈BH satisfying rA=A. Now, an application of Lemma 2 together with wS=wS∗ and wαS=∣α∣wS yields
(26)Hθ2≤14wΔf,gS2+124f∣S∣2Δf,gS∗g∣S∣2Δf,gS+SS∗+S∗S≤14wΔf,gS2+124fS2Δf,gS∗gS2Δf,gS+SS∗+S∗S=14wΔf,gS2+f∣S∣g∣S∣Δf,gS+SS∗+S∗S.
The last equality holds by using the factS=S∗. Now, we take supremum over θ∈ℝ to get
(27)supθ∈ℝHθ2≤supθ∈ℝ14wΔf,gS2+gSΔf,gSfS+SS∗+S∗S.
By using Lemma1 in above inequality, we obtain
(28)w2S≤14wΔf,gS2+gSΔf,gSfS+SS∗+S∗S,as required.The following inequality is another generalized bound of numerical radius.Theorem 5.
LetS∈BH. Then we have
(29)w4S≤116wΔf,gS2+g∣S∣Δf,gSf∣S∣2+18wS2P+PS2+116P2,where P=S∗S+SS∗.Proof.
Since(30)Hθ=12eιθS+e−ιθS∗forallθ∈ℝ,therefore,
(31)Hθ2=14eιθS+e−ιθS∗2=14e2ιθS2+e−2ιθS∗2+SS∗+S∗SHθ4=116e2ιθS2+e−2ιθS∗2+P2=116e2ιθS2+e−2ιθS∗22+e2ιθS2+e−2ιθS∗2P+Pe2ιθS2+e−2ιθS∗2+P2=116e2ιθS2+e−2ιθS∗22+e2ιθS2P+e−2ιθS∗2P+Pe2ιθS2+Pe−2ιθS∗2+P2=116e2ιθS2+e−2ιθS∗22+e2ιθS2P+PS2+e−2ιθS∗2P+PS∗2+P2=116e2ιθS2+e−2ιθS∗22+2Ree2ιθS2P+PS2+P2,where
(32)Ree2ιθS2P+PS2=e2ιθS2P+PS2+e−2ιθS2P+PS2∗2.
In third equalityP=S∗S+SS∗. Now, by using the properties of operator norm ∥·∥ on BH, we have
(33)Hθ4≤116e2ιθS2+e−2ιθS∗22+2Ree2ιθS2P+PS2+P2=116e2ιθUSUS+e−2ιθSU∗SU∗2+2Ree2ιθS2P+PS2+P2=116e2ιθUgSΔf,gSfS+e−2ιθfSΔf,gS∗gSU∗2+2Ree2ιθS2P+PS2+P2=116r2e2ιθUgSΔf,gSfS+e−2ιθfSΔf,gS∗gSU∗+2Ree2ιθS2P+PS2+P2=116r2M1N1+M2N2+2Ree2ιθS2P+PS2+P2,where M1=eιθUg∣S∣Δf,gS, N1=f∣S∣, M2=e−2ιθf∣S∣Δf,gS∗, and N2=g∣S∣U∗. The first equality obtained by using S=U∣S∣ and S∗=∣S∣U∗ in first inequality, the second equality obtained by using f∣S∣g∣S∣=∣S∣ and g∣S∣f∣S∣=∣S∣ in third equality, and the fifth equality holds for hermitian operator satisfying rA=∥A∥. Now, by using Lemma 2 together with wS=wS∗ and wαS=∣α∣wS, it yields
(34)Hθ4≤116wΔf,gS2+124f∣S∣2e−2ιθΔf,gS∗gS2eιθΔf,gS2+2Ree2ιθS2P+PS2+P2≤116wΔf,gS2+124fS2Δf,gS∗gS2Δf,gS2+2Ree2ιθS2P+PS2+P2=116wΔf,gS2+fSgSΔf,gS2+2Ree2ιθS2P+PS2+P2.
The last equality holds by using the factS=S∗. Now, we take supremum over θ∈ℝ to get
(35)supθ∈ℝHθ4≤supθ∈ℝ116wΔf,gS2+fSgSΔf,gS2+2Ree2ιθS2P+PS2+P2.
Applying Lemma1 on above inequality, we obtain
(36)w4S≤116wΔf,gS2+f∣S∣g∣S∣Δf,gS2+18wS2P+PS2+116P2.as required.To give the next bound of numerical radius, first, we define iterated generalized Aluthge transform. The iterated generalized Aluthge transform is defined as(37)Δf,gkS=ΔΔf,gk−1S;∀k∈ℕ,where f and g both are nonnegative and continuous functions.By using Theorem4 repeatedly, we can obtain numerical radius bound in terms of iterated generalized Aluthge transform.Theorem 6.
LetS∈BH be such that the sequence ∥Δf,gnS∥n=1∞ is convergent then
(38)w2S≤∑k=1∞14kf∣Δf,gk−1S∣g∣Δf,gk−1S∣Δf,gkS+Δf,gk−1S∗Δf,gk−1S+Δf,gk−1SΔf,gk−1S∗.Proof.
In order to prove the theorem, it is sufficient to prove the following assertion(39)w2S≤∑k=1n14kfΔf,gk−1SΔf,gkSgΔf,gk−1S+Δf,gk−1S∗Δf,gk−1S+Δf,gk−1SΔf,gk−1S∗+14nw2Δf,gnSforalln∈ℕ.
We use mathematical induction to prove the above assertion. An application of Theorem4 gives
(40)w2S≤14fSgSΔf,gS+S∗S+SS∗+14wΔf,gS2.
The use of the inequalitywS2≤w2S gives
(41)w2S≤14fSΔf,gSgS+S∗S+SS∗+14w2Δf,gS.
Thus the preliminary induction step holds. Now, suppose that(42)w2S≤∑k=1m14kfΔf,gk−1SΔf,gkSgΔf,gk−1S+Δf,gk−1S∗Δf,gk−1S+Δf,gk−1SΔf,gk−1S∗+14mw2Δf,gmSforsomem∈ℕ.
Then, another application of Theorem4 yields
(43)w2S≤∑k=1m14k∥f∣Δf,gkS∣∥∥Δf,gk+1S∥∥gΔf,gkS∥+Δf,gkS∗Δf,gkS+Δf,gkSΔf,gkS∗+14∥f∣Δf,gmS∣∥∥Δf,gm+1S∥∥gΔf,gmS∥+Δf,gmS∗Δf,gmS+Δf,gmSΔf,gmS∗+14m+1wΔf,gm+1S2.
Simplifying and using the inequalitywS2≤w2S gives
(44)w2S≤∑k=1m+114kfΔf,gkSΔf,gk+1SgΔf,gkS+Δf,gkS∗Δf,gkS+Δf,gkSΔf,gkS∗+14m+1w2Δf,gm+1S.
Hence, the assertion (39) holds for all n∈ℕ.
Now, using the inequalitywS≤∥S∥ in (39) and then using the hypothesis, we get the desired inequality (38).Remark 7.
It is easy to observe from the Theorems3–6 that the upper bounds (17)–(38) are generalized bounds. Indeed, if we take f∣S∣=Sλ and g∣S∣=S1−λ for λ∈0,1, in the bounds (17)–(38), then we obtain bounds (9)–(12).


## 3. Examples

In this section, we shall consider some choices off and g in generalized Aluthge transform (13) and use them to compute upper bounds of numerical radius for some matrices.Example 8.
GivenS=050001020. Then, S=U∣S∣ is a polar decomposition of S, where ∣S∣=0000290001, and U=05/29000102/290 is partial isometry. The bounds (14), (17), (22), and (29) are computed for some choices of f and g in (13) for given S in Table 1.Table 1
Bounds (14), (17), (22), and (29) for different choices of f and g in (13).
f,gBound (14)Bound (17)Bound (22)Bound (29)eS,Se−S22.728856.678740.174340.0320eS1/2,Se−S1/23.76954.19343.62383.2146S1/2,S1/23.48813.70783.33533.0645eS1/3,Se−S1/33.34683.63543.27073.0389The numerical radius ofS is
(45)wS=2.9154.Example 9.
LetS=030000022. Then, S=U∣S∣ is a polar decomposition of S, where ∣S∣=000030002, and U=010000001 is partial isometry.The bounds (14), (17), (22), and (29) are computed for some choices of f and g in (13) for given S in Table 2.Table 2
Bounds (14), (17), (22), and (29) for different choices of f and g in (13).
f,gBound (14)Bound (17)Bound (22)Bound (29)S1/3,S2/32.622452.370079432.278936152.1589862S1/2,S1/22.52.29128782.17944942.0963298eS1/3,Se−S1/32.52.291208152.17940752.09630510eS1/2,Se−S1/22.52.291205472.179406172.09630427The numerical radius ofS is
(46)wS=2.


## 4. Conclusion

Summarizing the investigation carried out, we note that generalized Aluthge transform (13) with additional conditions (i) and (ii) is useful in achieving the generalized upper bounds for numerical radius. It is proved in Theorems 3, Theorem 4, Theorem 5, and Theorem 6 that bounds (17), (22), (29), and (38) are upper bounds of numerical radius that generalize the upper bounds (9), (10), (11), and (12) of numerical radius already existing in the literature. Theoretical investigations are supported by examples in which computations are carried out for finding bounds (14), (17), (22), and (29) of numerical radius for some choices of the pair f,g in the generalized Aluthge transform Δf,g. Examples 8 and 9 demonstrate that generalized Aluthge transform provides a wide range of transforms that may be used as a tool to compute the upper bounds for numerical radius. These results might be helpful in studying perturbation, convergence, iterative solution methods, and integrative methods, which is the subject of future work. In the future, we also have a plan to investigate the lower bounds of numerical radius.



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