# On Some Numerical Radius Inequalities Involving Generalized Aluthge Transform

Authors: Javariya Hyder; Muhammad Saeed Akram; Abdulaziz S. Alofi; Dilruba Akter
Journal: Journal of Function Spaces
 (2022)

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## Abstract

LetS be any bounded linear operator defined on a complex Hilbert space H. In this paper, we present some numerical radius inequalities involving the generalized Aluthge transform to attain upper bounds for numerical radius. Numerical computations are carried out for some particular cases of generalized Aluthge transform.


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## Body

## 1. Introduction

Mathematical inequalities play an essential role in developing various areas of pure and applied mathematics. The usefulness of mathematical inequalities is to estimate the solutions of real-life problems in engineering and other fields of science. In mathematics, particularly in functional analysis, the study of numerical radius inequalities has become the attention of many researchers due to the applications of numerical radius in operator theory and numerical analysis, etc. (see [1–4]). Various mathematicians have developed number of numerical radius inequalities to estimate the upper and lower bounds for numerical radius. It is interesting for researchers to get the refinements and generalization of these inequalities. The aim of this paper is to study the generalization and refinements of existing inequalities for numerical radius. Now, we recall some notions to proceed our work.LetBH be the C∗-algebra of all bounded linear operators defined on a complex Hilbert space H. For S∈BH, the usual operator norm is defined as
(1)S=supSx:x∈H,x=1,and the numerical radius is defined as
(2)wS=supSx,x:x∈H,x=1.It is well known that the numerical radius defines an equivalent operator norm onBH, and for S∈BH, we have
(3)12S≤wS≤S.Many authors worked on the refinement of inequality (3) (see [5–7]). Kittaneh developed the following upper bound of numerical radius:
(4)wS≤12S+S21/2,which is a refinement of inequality (3) (see [5]). For S∈BH having polar decomposition S=US where U is a partial isometry and S=S∗S1/2, the Aluthge transform is defined as
(5)S~=S1/2US1/2,see [8]. Yamazaki developed an upper bound of numerical radius involving Aluthge transform given by
(6)wS≤12S+wS~,which is an improvement of bounds (3) and (4) (see [9]). Bhunia et al. developed a bound of the numerical radius given by
(7)w2S≤14S2+w2S~+wSS~+S~S,and proved that it is a refinement of bound (6) (see [10]). Okubo introduced a generalization of Aluthge transform which is defined as
(8)Sλ~=SλUS1−λ,for λ∈0,1, known as λ-Aluthge transform (see [11]). After that, a number of numerical radius inequalities were established involving λ-Aluthge transform (see [12–14]). Abu Omar and Kittaneh using λ-Aluthge transform generalized bound (6) given by
(9)wS≤12S+wSλ~,see [12]. Shebrawi and Bakherad introduced another generalization of Aluthge transform which is defined as
(10)Sf,g~=fSUgS,where f and g are nonnegative and continuous functions such that fxgx=xx≥0, known as generalized Aluthge transform. The authors generalized inequality (9) given by
(11)wS≤12S+wSf,g~,(12)wS≤14f2S+g2S+12wSf,g~,see [15].In this paper, we establish some new inequalities of the numerical radius using generalized Aluthge transform. Specifically, we generalize inequality (7) and improve the inequalities (3), (4), and (12). Some examples of operators are presented for which the bounds of numerical radius are computed from these inequalities for some choices of f,g in (10).


## 2. Main Results

Now, we recall a lemma that will be used to achieve our goals.Lemma 1 (see [9]).
LetS∈BH; then, for θ∈ℝ, we have
(13)wS=supθ∈ℝHθ=supθ∈ℝReeiθS,where Hθ=eiθS+e−iθS∗/2.Polarization identity: [15] For each x1,y1∈H, we have
(14)x1,y1=14x1+y12−x1−y12+ix1+iy12−ix1−iy12.Now, we establish an inequality of numerical radius which is a generalization of inequality (7) and a refinement of inequality (12).Theorem 2. LetS∈BH.
Then,(15)w2S≤116Q2+14w2Sf,g~+18wQSf,g~+Sf,g~Q,where Q=fS2+gS2 and f,g is nonnegative continuous functions defined on 0,∞ such that ftgt=t.Proof.
LetS=US be the polar decomposition of S. Then, by polarization identity, we have
(16)eiθSx,x=eiθUSx,x=eiθUgSfSx,x=eiθfSx,gSU∗x=14e2iθfSx+gSU∗x2−e2iθfSx−gSU∗x2+ie2iθfSx+igSU∗x2−ie2iθfSx−igSU∗x2.
Therefore,(17)ReeiθSx,x=14e2iθfS+gSU∗x2−14e2iθfS−gSU∗x2≤14e2iθfS+gSU∗2=14e2iθfS+gSU∗e2iθfS+gSU∗∗=14e2iθfS+gSU∗e−2iθfS+UgS=14fS2+gS2+e2iθfSUgS+e−2iθgSU∗fS=14fS2+gS2+e2iθSf,g~+e−2iθSf,g~∗=14Q+2Ree2iθSf,g~=14Q+2Ree2iθSf,g~21/2=14Q2+4Ree2iθSf,g~2+2QRee2iθSf,g~+2Ree2iθSf,g~Q1/2=14Q2+4Ree2iθSf,g~2+2Ree2iθQSf,g~+Sf,g~Q1/2≤14Q2+4Ree2iθSf,g~2+2Ree2iθQSf,g~+Sf,g~Q1/2.
Now taking supremum overθ∈ℝ in the last inequality and then applying Lemma 1, we obtain
(18)w2S≤116Q2+14w2Sf,g~+18wQSf,g~+Sf,g~Q,as desired.Theorem2 includes some particular cases of generalized Aluthge transform for different choices of continuous functions f and g in (10) as follows.Corollary 3. LetS∈BH.
Then, forλ∈0,1, we have
(19)w2S≤116Q2+14w2Se~+18wQSe~+Se~Q,where Q=eSλ2+Se−Sλ2 and Se~=eSλUSeS−λ.Corollary 4. LetS∈BH.
Then, forλ∈0,1, we have
(20)w2S≤116Q2+14w2Se~+18wQSe~+Se~Q,where Q=Se−Sλ2+eSλ2 and Se~=SeS−λUeSλ.Corollary 5. LetS∈BH.
Then, forλ∈0,1, we have
(21)w2S≤116Q2+14w2Sλ~+18wQSλ~+Sλ~Q,where Q=S2λ+S21−λ. In particular,
(22)w2S≤14S2+w2S~+wSS~+S~S.Remark 6.
By using the inequality(23)wYA+AY∗≤2YwA,for all Y,A∈BH (see [16]) in inequality (15) obtained in Theorem 2, we have
(24)w2S≤116Q2+14w2Sf,g~+182QwSf,g~=14Q+12wSf,g~2.Hence,(25)wS≤14Q+12wSf,g~,where Q=fS2+gS2. Thus, inequality (15) obtained in Theorem 2 is better than inequality (12).Remark 7.
For continuous functionsf and g in (10), if Sf,g~=0, then inequality (15) becomes
(26)wS≤14fS2+gS2.In particular, if we takefS=S1/2 and gS=S1/2, for this choice of f and g if Sf,g~=0, then inequality (15) becomes wS≤1/2S, and combined with inequality (3), we get wS=1/2S.Theorem 8. LetS∈BH.
Then, we have(27)w4S≤14fSSf,g~gS2+18wS2P+PS2+116P2,where P=S∗S+SS∗ and f,g is nonnegative continuous functions defined on 0,∞ such that ftgt=t.Proof.
SinceHθ=eiθS+e−iθS∗/2 for all θ∈ℝ, then we have
(28)Hθ2=14e2iθS2+e−2iθS∗2+SS∗+S∗S=14e2iθS2+e−2iθS∗2+P,which yields
(29)Hθ4=116e2iθS2+e−2iθS∗22+e2iθS2P+PS2+e−2iθS∗2P+PS∗2+P2=116e2iθS2+e−2iθS∗22+2Ree2iθS2P+PS2+P2=116e2iθUSUS+e−2iθSU∗SU∗2+2Ree2iθS2P+PS2+P2=116e2iθUgSfSUgSfS+e−2iθfSgSU∗fSgSU∗2+2Ree2iθS2P+PS2+P2=116e2iθUgSSf,g~fS+e−2iθfSSf,g~∗gSU∗2+2Ree2iθS2P+PS2+P2.SinceHθ is self-adjoint, so Hθ4=Hθ4. Hence, using the properties of operator norm · on BH, we have
(30)Hθ4≤116e2iθUgSSf,g~fS+e−2iθfSSf,g~∗gSU∗2+2Ree2iθS2P+PS2+P2≤116UgSSf,g~fS+fSSf,g~∗gSU∗2+2Ree2iθS2P+PS2+P2≤116UgSSf,g~fS+fSSf,g~∗gSU∗2+2Ree2iθS2P+PS2+P2=1164fSSf,g~gS2+2Ree2iθS2P+PS2+P2,where the equality holds because S=S∗. Now taking supremum over θ∈ℝ in last equality, then applying Lemma 1, yields
(31)w4S≤14fSSf,g~gS2+18wS2P+PS2+116P.For different choices off and g in (10), we obtain the following inequalities of numerical radius from Theorem 8.Corollary 9. LetS∈BH.
Then, forλ∈0,1, we have
(32)w4S≤14eSλSe~Se−Sλ2+18wS2P+PS2+116P2,where P=S∗S+SS∗ and Se~=eSλUSe−Sλ.Corollary 10. LetS∈BH.
Then, forλ∈0,1, we have
(33)w4S≤14eSλSe~Se−Sλ2+18wS2P+PS2+116P2,where P=S∗S+SS∗ and Se~=SeS−λUeSλ.Corollary 11. LetS∈BH.
Then, forλ∈0,1, we have
(34)w4S≤14SSλ~2+18wS2P+PS2+116P2,where P=S∗S+SS∗. In particular,
(35)w4S≤14SS~2+18wS2P+PS2+116P2.Remark 12.
It is easy to check thatS~≤S21/2 (see [9] for details). Using the following inequality
(36)wYA+AY∗≤2YwA,for all Y,A∈BH (see [14]), Corollary 11 yields
(37)w4S≤14SS~2+18wS2P+PS2+116P2≤14SS21/22+14wS2P+116P2≤14SS21/22+14S2P+116P2=12SS21/2+14P2.We know thatS∗S+SS∗≤S2+S2 (see [17]). Hence,
(38)12SS21/2+14S∗S+SS∗≤12SS21/2+14S2+14S2=12S21/2+12S2.Thus, the bound given in Corollary11 is better than bound (4).Remark 13.
IfSf,g~=0 in inequality (27) obtained in Theorem 8 for different choices of f and g in (10), then inequality (27) becomes
(39)w4S≤18wS2P+PS2+116P2.IfS2=0 and Sf,g~=0 are equivalent conditions, then inequality (27) becomes
(40)w2S=14P.Theorem 14. LetS∈BH.
Then, we have(41)w3S≤14fSSf,g~2gS+wS2S∗+S∗S2+SS∗S,where f,g is nonnegative continuous functions defined on 0,∞ such that ftgt=t.Proof.
SinceHθ=eiθS+e−iθS∗/2 for all θ∈ℝ, then we have
(42)Hθ2=14e2iθS2+e−2iθS∗2+SS∗+S∗S,which implies
(43)Hθ3=18eiθS+e−iθS∗e2iθS2+e−2iθS∗2+SS∗+S∗S=18e3iθS3+e−3iθS∗3+2ReeiθS2S∗+S∗S2+SS∗S.In the last equality,ReeiθS2S∗+S∗S2+SS∗S=eiθS∗S2+S∗2S+S∗SS∗+e−iθS2S∗+S∗S2+SS∗S/2. Hence,
(44)Hθ3≤18e3iθUgSSf,g~2fS+e−3iθfSSf,g~∗2gSU∗+2ReeiθS2S∗+S∗S2+SS∗S≤18UgSSf,g~2fS+fSSf,g~∗2gSU∗+2ReeiθS2S∗+S∗S2+SS∗S=14fSSf,g~2gS+ReeiθS2S∗+S∗S2+SS∗S.The first inequality holds becauseA1+A2≤A1+A2 where A,A1,A2∈BH, the second inequality holds because A1A2≤A1A2 and An≤An∀n∈ℕ, and the third equality holds because A=A∗ and U=U∗=1. Now taking supremum over θ∈ℝ in above equality the using Lemma 1, we obtain
(45)w3S≤14fSSf,g~2gS+wS2S∗+S∗S2+SS∗S,as desired.Corollary 15. LetS∈BH.
Then, forλ∈0,1, we have
(46)w3S≤14eSλSe~2Se−Sλ+wS2S∗+S∗S2+SS∗S,where Se~=eSλUSe−Sλ.Corollary 16. LetS∈BH.
Then, forλ∈0,1, we have
(47)w3S≤14eSλSe~2Se−Sλ+wS2S∗+S∗S2+SS∗S,where Se~=SeS−λUeSλ.Corollary 17. LetS∈BH.
Then, forλ∈0,1, we have
(48)w3S≤14SSλ~2+wS2S∗+S∗S2+SS∗S.
In particular,(49)w3S≤14SS~2+wS2S∗+S∗S2+SS∗S.Remark 18.
Yan et al. proved that(50)w2S≤12fSSf,g~gS+14S∗S+SS∗,see [18]. Inequality (41) obtained in Theorem 14 gives better bounds of numerical radius of S for different choices of f and g in (10) when
(51)S=050003100.
Then,S=US is a polar decomposition of S, where
(52)S=000052003,and
(53)U=010001100is partial isometry.Bounds (41) and (50) are computed for some choices of f and g in (10) for the given S in Table 1, whereas the numerical radius of S is
(54)wS=2.9154.Table 1
Bounds (41) and (50) for different choices of f and g in (10).
f,gBound (50)Bound (41)S1/2,S1/24.26403.2655S1/7,S6/74.07043.0724eS1/10,Se−S1/104.35823.3622eS1/10,Se−S1/104.20553.2031The spectral radius of an operator is defined as(55)rS=supx=1λ:λ∈σS,where r denotes the spectral radius. For further inforamtion on spectral radius, see [19]. The following theorem will be used to develop the next inequality of numerical radius.Theorem 19 (see [17]).
LetM1,M2,N1,N2∈BH. Then,
(56)rM1N1+M2N2≤12wN1M1+wN2M2+12wN1M1−wN2M2+4N1M2N2M1.Theorem 20. LetS∈BH.
Then,(57)w3S≤18wSf,g~3+fSSf,g~2gS+14wS2S∗+S∗S2+SS∗S,where f,g is nonnegative continuous functions defined on 0,∞ such that ftgt=t.Proof.
SinceHθ=eiθS+e−iθS∗/2 for all θ∈ℝ, then we have
(58)Hθ2=14e2iθS2+e−2iθS∗2+SS∗+S∗S,which implies
(59)Hθ3=18eiθS+e−iθS∗e2iθS2+e−2iθS∗2+SS∗+S∗SHθ3=18e3iθUgSSf,g~2fS+e−3iθfSSf,g~∗2gSU∗+2ReeiθS2S∗+S∗S2+SS∗S.Now by using the properties of operator norm· on BH, we have
(60)Hθ3≤18e3iθUgSSf,g~2fS+e−3iθfSSf,g~∗2gSU∗+2ReeiθS2S∗+S∗S2+SS∗S=18re3iθUgSSf,g~2fS+e−3iθfSSf,g~∗2gSU∗+2ReeiθS2S∗+S∗S2+SS∗S=18rM1N1+M2N2+2ReeiθS2S∗+S∗S2+SS∗S,where M1=e3iθUgSSf,g~2,N1=fS,M2=e−3iθfSSf,g~∗2, and N2=gSU∗; the first equality holds for Hermitian operator satisfying rA=A. Now applying Theorem 19 on last equality with wαA=αwA and wA=wA∗, we obtain
(61)Hθ3≤18wSf,g~3+124fS2Sf,g~∗2gS2Sf,g~2+2ReeiθS2S∗+S∗S2+SS∗S≤18wSf,g~3+124fS2Sf,g~∗2gS2Sf,g~2+2ReeiθS2S∗+S∗S2+SS∗S≤18wSf,g~3+fSSf,g~2gS+14ReeiθS2S∗+S∗S2+SS∗S.Taking supremum overθ∈ℝ in last inequality, then applying Lemma 1, we obtain
(62)w3S≤18wSf,g~3+fSSf,g~2gS+14wS2S∗+S∗S2+SS∗S.Corollary 21. LetS∈BH.
Then, forλ∈0,1 we have
(63)w3S≤18wSe~3+eSλSe~2Se−Sλ+14wS2S∗+S∗S2+SS∗S,where Se~=eSλUSe−Sλ.Corollary 22. LetS∈BH.
Then, forλ∈0,1 we have
(64)w3S≤18wSe~3+eSλSe~2Se−Sλ+14wS2S∗+S∗S2+SS∗S,where Se~=SeS−λUeSλ.Corollary 23. LetS∈BH.
Then, forλ∈0,1 we have
(65)w3S≤18wSλ~3+SSλ~2+14wS2S∗+S∗S2+SS∗S.
In particular,(66)w3S≤18wS~3+SS~2+14wS2S∗+S∗S2+SS∗S.Remark 24.
It is easy to observe that inequality (57) obtained in Theorem 20 is better than inequality (41).
Now, we exhibit some examples where numerical radius bounds are computed from inequalities (15), (27), (41), and (57) for some choices of pair f,g in (10) and for a given operator S.Example 1.
Given(67)S=030002100.
Then,S=US is a polar decomposition of S, where
(68)S=100030002,and
(69)U=010001100is partial isometry.Bounds (15), (27), (41), and (57) are computed for some choices of f and g in (10) for the given S in Table 2, whereas the numerical radius of S is
(70)wS=2.0565.Table 2
Bounds (15), (27), (41), and (57) for different choices of f and g in (10).
f,gBound (15)Bound (27)Bound (41)Bound (57)S1/2,S1/22.27042.35962.46622.3811S1/3,S2/32.35962.32642.43462.3332S1/4,S3/42.48172.32442.43272.3632S1/5,S4/52.58262.35102.45792.3462eS1/2,Se−S1/28.72232.42342.52842.3854eS1/3,Se−S1/35.33892.34922.45632.3459eS1/4,Se−S1/44.43072.32772.43112.3314eS1/5,Se−S1/54.03332.33802.44562.3394Example 2.
Let(71)S=010002100.
Then,S=US is a polar decomposition of S, where
(72)S=100010002,and
(73)U=010001100.Bounds (15), (27), (41), and (57) are computed for some choices of f and g in (10) for the given S in Table 3, whereas the numerical radius of S is
(74)wS=1.3662.Table 3
Bounds (15), (27), (41), and (57) for different choices of f and g in (10).
f,gBound (15)Bound (27)Bound (41)Bound (57)S1/2,S1/21.50471.50651.45001.3880S1/3,S2/31.54161.54251.49011.4059S1/4,S3/41.57541.56371.51301.4432S1/5,S4/51.60261.57691.52761.4513eS1/2,Se−S1/24.73361.52721.47261.4004eS1/3,Se−S1/33.69231.53351.47941.4250eS1/4,Se−S1/43.33181.55851.50691.4399eS1/5,Se−S1/53.15291.57291.52311.4488


## 3. Conclusion

From the results of this paper, we conclude that the inequalities of numerical radius involving generalized Aluthge transform have variety of upper bounds for numerical radius due to the choice off,g in generalized Aluthge transform (10). The inequalities (15), (27), (41) and (57) obtained in Theorem 2, Theorem 8, Theorem 20, and Theorem 14 are new and generalized upper bounds for numerical radius. These generalized upper bounds can be useful to find better bounds of numerical radius already existing in literature for some choices of f,g in generalized Aluthge transform (10) and certain operators. It is proved that inequality (15) of Theorem 2 generalizes inequality (7) and improves inequality (12) for any choice of f,g in (10). Inequality (27) of Theorem 8 is sharper than inequality (12) for the choice of ft=gt=t1/2 in (10). Inequality (41) of Theorem 14 is better than inequality (57) of Theorem 20. But for inequality (57) of Theorem 20, we can find such matrix and pairs of f,g for which the inequality of Theorem 20 can give better bound of numerical radius available in literature. To support theoretical investigations, some examples are presented where numerical radius and its upper bounds are computed for the pairs f,g in generalized Aluthge transform. Examples 1 and 2 show that there is no comparison between the bounds obtained from the inequalities (15), (27), and (41) of Theorem 2, Theorem 8, and Theorem 20; however, generalized Aluthge transform has choices of the pair f,g in (10) for which better upper bounds can be computed for certain operators.



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