# Some Singular Value Inequalities for Sector Matrices Involving Operator Concave Functions

Authors: Chaojun Yang
Journal: Journal of Mathematics
 (2022)

---

## Abstract

In this paper, we give some singular value inequalities for sector matrices involving operator concave function, which are generalizations of some existing results. Moreover, we present some unitarily invariant norm inequalities for sector matrices.


---

## Body

## 1. Introduction

Throughout this paper, letMnℂ be the set of all n×n complex matrices. We denote by In the identity matrix in Mnℂ. For A∈Mnℂ, the conjugate transpose of A is denoted by A∗, and the matrices ℜA=1/2A+A∗ and ℑA=1/2iA−A∗ are called the real part and imaginary part of A, respectively (see [1], p. 6 and [2], p. 7). Moreover, A is called accretive if ℜA>0. For two Hermitian matrices A,B∈Mnℂ, we write A≥B (resp., A>B) if A−B is positive semidefinite (resp., positive definite).Matrix inequalities have many important applications in solving matrix equations; for more information, we refer the readers to see [3, 4]. In this paper, we mainly deal with matrix singular value inequalities.A real valued continuous functionf on an interval J is called matrix concave of order n if fαA+1−αB≥αfA+1−αfB for any two Hermitian matrices A,B∈Mn with spectrum in J and all α∈0,1. Furthermore, f is called operator concave if f is matrix concave for all n. We also note that f on interval J is operator monotone if for any two positive semidefinite A,B∈Mnℂ with spectra in J, A≥B implies fA≥fB. It is well-known that f is operator monotone if and only if f is operator concave.Let·u denote any unitarily invariant norm on Mnℂ, which satisfies UAVu=Au for any unitary matrices U,V∈Mnℂ and all A∈Mnℂ. Let s1A≥s2A≥⋯≥snA denote the singular values of A.The numerical range ofA∈Mnℂ is defined by(1)WA=x∗Ax:x∈ℂn,x∗x=1.Forα∈0,π/2, Sα denotes the sector region in the complex plane as follows:(2)Sα=z∈ℂ:ℜz>0,ℑz≤ℜztanα.IfWA⊆S0, then A is positive definite, and if WA,WB⊆Sα for some α∈0,π/2, then WA+B⊆Sα, A is nonsingular, and ℜA is positive definite. Moreover, WA⊆Sα implies WX∗AX⊆Sα for any nonzero n×m matrix X, thus WA−1⊆Sα. Recent developments on sector matrices can be found in [5–8].For two positive definite matricesA,B∈Mnℂ, the geometric mean, harmonic mean, and arithmetic mean are defined, respectively, as follows: A♯B=A1/2A−1/2BA−1/21/2A1/2, A!B=A−1+B−1/2−1, and A∇B=A+B/2. The arithmetic-geometric-harmonic mean inequalities for positive definite matrices A,B∈Mnℂ state that(3)A!B≤A♯B≤A∇B.For two accretive matricesA,B∈Mnℂ, Drury [7] defined the geometric mean of A and B as follows:(4)A♯B=2π∫0∞tA+t−1Bdtt−1.This new geometric mean defined by (4) possesses some similar properties compared with the geometric mean of positive matrices. For instance, A♯B=B♯A and A♯B−1=A−1♯B−1. Moreover, if A,B∈Mnℂ with WA,WB⊂Sα, then WA♯B⊂Sα.For the sake of convenience, we will need the following notation:(5)m=gx,whereg:0,∞⟶0,∞is an operator monotone function withg1=1.Forf∈m, by a limit argument, we know that limt⟶0+ft≥0. One may wonder what happens if limt⟶0+ft=∞? We note that limt⟶0+ft<∞. Otherwise, limt⟶0+ft=∞; since f is continuous and increasing, we can find x∈0,1 such that fx=∞>1=f1, a contradiction. Thus, limt⟶0+ft<∞. Throughout this paper, by function f:0,∞⟶0,∞ with f∈m, we mean g∈m and ft=gtt>0,limt⟶0+gtt=0. Thus, one can easily verify that f is operator monotone.Recently, Bedrani, Kittaneh, and Sababheh [5] characterized the operator monotone function for an accretive matrix: let A∈Mnℂ be accretive and f∈m,(6)fA=∫011−sI+sA−1−1dvfs,where vf is the probability measure satisfying fx=∫011−sI+sA−1−1dvfs.In [9], Garg and Aujla showed the following inequalities:(7)∏j=1ksjA+Br≤∏j=1ksjIn+|A|r∏j=1ksjIn+Brfor1≤k≤n,1≤r≤2,(8)∏j=1ksjIn+fA+B≤∏j=1ksjIn+fA∏j=1ksjIn+fBfor1≤k≤n,where A,B∈Mn and f:0,∞⟶0,∞ is an operator concave function.By lettingA,B≥0 in (7) and (8), we have(9)∏j=1ksjA+Br≤∏j=1ksjIn+Ar∏j=1ksjIn+Brfor1≤k≤n,1≤r≤2,(10)∏j=1ksjIn+fA+B≤∏j=1ksjIn+fA∏j=1ksjIn+fBfor1≤k≤n.Recently, Yang and Lu [10] obtained two singular value inequalities for sector matrices based on (7) and (8). Later, Lin and Fu [11], Xue and Hu [12], and Nasiri and Furuichi [13] independently gave some singular value inequalities for sector matrices related to Yang and Lu’s results.In this paper, we give some singular value inequalities for sector matrices based on (9) and (10), which are generalizations of some existing results mentioned above. Moreover, we present some unitarily invariant norm inequalities for sector matrices.


## 2. Main Results

We begin this section with some results from the literature which will be necessary for achieving our goals.Lemma 2.1.
(see [5]). Let A∈Mnℂ with WA⊆Sα. If f∈m, then(11)fℜA≤ℜfA≤sec2αfℜA.Lemma 2.2.
(see [1], p.73]). Let A∈Mnℂ be accretive. Then,(12)sjℜA≤sjA,j=1,2,…,n.Consequently,(13)ℜAu≤Au.Lemma 2.3.
(see [8]). Let A∈Mnℂ be such that WA⊆Sα. Then,(14)sjA≤sec2αsjℜA,j=1,2,…,n.Lemma 2.4.
(see [14]). Let A∈Mnℂ be such that WA⊆Sα for some α∈0,π/2. Then,(15)Au≤secαℜAu.Lemma 2.5.
(see [15]). Let A∈Mnℂ be such that WA⊆Sα for some α∈0,π/2. Let A=UA be the polar decomposition of A. Then,(16)A≤secα2ℜA+U∗ℜAU.Lemma 2.6.
(see [16–18]). Let A,B∈Mnℂ be such that WA,WB⊆Sα and f∈m. Then,(17)ℜA♯ℜB≤ℜA♯B≤sec2αℜA♯ℜB.The inequalities also hold if replacing♯ by !.The following lemma is a well-known result.Lemma 2.7.
(see [19]). If f:0,∞⟶0,∞ is operator monotone, then fαt≤αft for α≥1.Lemma 2.8.
(see [20]). If f:0,∞⟶0,∞ is monotone concave, then for positive semidefinite matrices A,B∈Mnℂ, there exist unitary matrices U,V∈Mnℂ such that fA+B≤U∗fAU+V∗fBV.Lemma 2.9.
(see [8, 21]). Let A,B∈Mnℂ with WA⊆Sα. Then,(18)ℜA−1≤ℜ−1A≤sec2αℜA−1.Theorem 2.10.
LetA,B∈Mnℂ be such that WA,WB⊆Sα and let ft=gtt>0,limt⟶0+gtt=0. for g∈m. Then,(19)∏j=1ksjIn+fA+B∏j=1ksjIn+sec2αfA∏j=1ksjIn+sec2αfB,where 1≤k≤n.Proof.
We have the following chain of inequalities:(20)∏j=1ksjIn+fA+B≤∏j=1ksjIn+fA∏j=1ksjIn+fBby4≤∏j=1ksjIn+fsec2αUℜAU∗∏j=1ksjIn+fsec2αUℜBU∗by Lemma2.3≤∏j=1ksjIn+fsec2αUℜAU∗∏j=1ksjIn+fsec2αUℜBU∗by Lemma2.7≤∏j=1ksjIn+fsec2αfℜA∏j=1ksjIn+fsec2αfℜB≤∏j=1ksjIn+fsec2αℜfA∏j=1ksjIn+fsec2αℜfBby Lemma2.1≤∏j=1ksjIn+fsec2αfA∏j=1ksjIn+fsec2αfBby Lemma2.2,where UandV are unitary matrices.
This completes the proof.Remark 2.11.
Since for some unitary matrixU, we have(21)∏j=1ksjIn+A+B=∏j=1ksjIn+A+B≤∏j=1ksjIn+UA+BU∗≤∏j=1ksjIn+A+B,(22)∏j=1ksjA+B=∏j=1ksjA+B≤∏j=1ksjIn+A+B.
By lettingft=t in Theorem 2.10, we obtain Theorem 2.8 in [11]; thus, Theorem 2.10 is a generalization of Theorem 2.8 in [11].Theorem 2.12.
LetA,B∈Mnℂ be such that WA,WB⊆Sα and let ft=gtt>0,limt⟶0+gtt=0, for g∈m. Then,(23)∏j=1ksjIn+fA+B≤∏j=1ksj2In+fsecα2A∏j=1ksj2In+fsecα2B,where 1≤k≤n.Proof.
We have the following chain of inequalities:(24)∏j=1ksjIn+fA+B≤∏j=1ksjIn+fA∏j=1ksjIn+fBby4≤∏j=1ksjIn+fsecα2ℜA+U1ℜAU1∗∏j=1ksjIn+fsecα2ℜB+U1ℜBU1∗by Lemma2.5≤∏j=1ksjIn+U2∗fsecα2ℜAU2+U3∗fsecα2ℜAU3×∏j=1ksjIn+V2∗fsecα2ℜBV2+V3∗fsecα2ℜBV3by Lemma2.8≤∏j=1ksjIn+U2∗fsecα2ℜAU2∏j=1ksjIn+U3∗fsecα2ℜAU3×∏j=1ksjIn+V2∗fsecα2ℜBV2∏j=1ksjIn+V3∗fsecα2ℜBV3by6=∏j=1ksj2In+fsecα2ℜA∏j=1ksj2In+fsecα2ℜB≤∏j=1ksj2In+ℜfsecα2A∏j=1ksj2In+ℜfsecα2Bby Lemma2.1≤∏j=1ksj2In+fsecα2A∏j=1ksj2In+fsecα2Bby Lemma2.2,where U1,U2,U3,V1,V2,andV3 are unitary matrices.
This completes the proof.Remark 2.13.
By lettingft=t in Theorem 2.12 and taking Remark 2.11 into consideration, we note that Theorem 2.12 is a generalization of Theorem 2.6 in [12].Theorem 2.14.
LetA,B∈Mnℂ be such that WA,WB⊆Sα. Then,(25)∏j=1ksjA+Br≤sec2kα∏j=1ksj2In+Ar/2∏j=1ksj2In+Br/2,where 1≤k≤n and 1≤r≤2.Proof.
For positive semidefinite matrixS∈Mnℂ, we have In+S1/22≥In+S, which is equivalent to S≥0. Thus, we obtain(26)cos2kα∏j=1ksjA+Br≤∏j=1ksjℜA+Brby Lemma2.3≤∏j=1ksjℜA+Brby Theorem3.5in6=∏j=1ksjℜA+ℜBr≤∏j=1ksjIn+ℜAr∏j=1ksjIn+ℜBrby5≤∏j=1ksj2In+ℜAr/2∏j=1ksj2In+ℜBr/2by Lemma2.1≤∏j=1ksj2In+Ar/2∏j=1ksj2In+Br/2by Lemma2.2,which completes the proof.Theorem 2.15.
LetA,B∈Mnℂ be such that WA,WB⊆Sα. Then,(27)∏j=1ksjA+B−1≤sec2kα4k∏j=1ksjIn+sec2αA−1∏j=1ksjIn+sec2αB−1,where 1≤k≤n.Proof.
By inequality (3), we obtain(28)ℜA+ℜB−1≤14ℜ−1A+ℜ−1B.
We have the following chain of inequalities:(29)∏j=1ksjA+B−1≤sec2kα∏j=1ksjℜA+B−1by Lemma2.3≤sec2kα∏j=1ksjℜA+B−1by Lemma2.9=sec2kα∏j=1ksjℜA+ℜB−1≤sec2kα4k∏j=1ksjℜ−1A+ℜ−1Bby8≤sec2kα4k∏j=1ksjIn+ℜ−1A∏j=1ksjIn+ℜ−1Bby Lemma2.3≤sec2kα4k∏j=1ksjIn+sec2αℜA−1∏j=1ksjIn+sec2αℜB−1by Lemma2.9≤sec2kα4k∏j=1ksjIn+sec2αA−1∏j=1ksjIn+sec2αB−1by Lemma2.2,which completes the proof.Example 2.16.
Let(30)A−1=152128−152128i00152128−152128i,(31)B−1=92128−92128i0092128−92128i.
We compute the right hand side of (27) and (2.7) in [13], respectively, as follows:(32)sec4π/442∏j=12sjI2+sec2π4A−1∏j=12sjI2+sec2π4B−1=0.6952,sec8π/442∏j=12sjI2+A−1∏j=12sjI2+B−1=1.6893.
This example shows that inequality (27) is stronger than inequality (2.7) in [13].Example 2.17.
Let(33)A−1=932128−292128i00932128−292128i,(34)B−1=812128−392128i00812128−392128i.
We compute the right hand side of (27) and (2.7) in [13], respectively, as follows:(35)sec4π/442∏j=12sjI2+sec2π4A−1∏j=12sjI2+sec2π4B−1=20.7684,sec8π/442∏j=12sjI2+A−1∏j=12sjI2+B−1=15.9122.
This implies that inequality (27) is weaker than inequality (2.7) in [13].Theorem 2.18.
LetA,B∈Mnℂ be such that WA,WB⊆Sα and let ft=gtt>0,limt⟶0+tt=0. for g∈m. Then,(36)∏j=1ksjIn+fA+B≤sec2kα∏j=1ksjIn+sec2αfA∏j=1ksjIn+sec2αfB,(37)∏j=1ksjfA+B≤sec4kα∏j=1ksjIn+fA∏j=1ksjIn+fB,where 1≤k≤n.Proof.
Compute(38)∏j=1ksjIn+fA+B≤sec2kα∏j=1ksjIn+ℜfA+Bby Lemma2.3≤sec2kα∏j=1ksjIn+sec2αfℜA+ℜBby Lemma2.1≤sec2kα∏j=1ksjIn+sec2αfℜA∏j=1ksjIn+sec2αfℜBby6≤sec2kα∏j=1ksjIn+sec2αℜfA∏j=1ksjIn+sec2αℜfBby Lemma2.1≤sec2kα∏j=1ksjIn+sec2αfA∏j=1ksjIn+sec2αfBby Lemma2.2.
To prove (37), compute(39)∏j=1ksjfA+B≤sec2kα∏j=1ksjℜfA+Bby Lemma2.3≤sec4kα∏j=1ksjfℜA+ℜBby Lemma2.1≤sec4kα∏j=1ksjIn+fℜA∏j=1ksjIn+fℜBby6≤sec4kα∏j=1ksjIn+ℜfA∏j=1ksjIn+ℜfBby Lemma2.1≤sec4kα∏j=1ksjIn+fA∏j=1ksjIn+fBby Lemma2.1,which completes the proof.Theorem 2.19.
LetA,B∈Mnℂ be such that WA,WB⊆Sα and let ft=gtt>0,limt⟶0+tt=0. for g∈m. Then,(40)In+fA+Bu≤secαIn+sec2αfAuIn+sec2αfBu,(41)fA+Bu≤sec3αIn+fAuIn+fBu.Proof.
LetS,T∈Mnℂ be positive semidefinite. By inequality (6), we obtain(42)∏j=1ksj1/2In+fS+T≤∏j=1ksj1/2In+fS∏j=1ksj1/2In+fT.
By the property that weak log-majorization implies weak majorization and Cauchy-Schwarz inequality, we get(43)∑j=1ksj1/2In+fS+T≤∑j=1ksjIn+fS1/2∑j=1ksjIn+fT1/2,which is equivalent to(44)In+fS+T1/2k2≤In+fSkIn+fTk.
According to Fan’s dominance principle (see [1], p.93), we have(45)In+fS+T1/2u2≤In+fSuIn+fTu.
Hence,(46)In+fS+Tu=In+fS+Tu=In+fS+T1/22u≤In+fS+T1/22u2≤In+fSuIn+fTu.
Now, we have(47)In+fA+Bu≤secαIn+ℜfA+Buby Lemma2.4≤secαIn+fℜA+ℜBuby Lemma2.1≤secαIn+fℜAusecαIn+fℜBuby11≤secαIn+ℜfAusecαIn+ℜfBuby Lemma2.1≤secαIn+fAusecαIn+fBuby Lemma2.2.
To prove (41), we compute(48)In+fA+Bu≤secαIn+ℜfA+Buby Lemma2.4≤sec3αIn+fℜA+ℜBuby Lemma2.1≤sec3αIn+fℜAusecαIn+fℜBuby11≤sec3αIn+ℜfAusecαIn+ℜfBuby Lemma2.1≤sec3αIn+fAusecαIn+fBuby Lemma2.2,which completes the proof.Theorem 2.20.
LetA,B∈Mnℂ be such that WA,WB⊆Sα. Then,(49)A♯Bu≤sec3α2In+AuIn+Bu.Proof.
We have the following chain of inequalities:(50)A♯Bu≤secαℜA♯Buby Lemma2.4≤sec3αℜA♯ℜBuby Lemma2.6≤sec3α2ℜA+ℜBuby1≤sec3α2In+ℜAuIn+ℜBuby11≤sec3α2In+AuIn+Buby Lemma2.2,which completes the proof.Remark 2.21.
We remark that Theorem 2.20 is a refinement of Proposition 3.2 in [22].Theorem 2.22.
LetA,B∈Mnℂ be such that WA,WB⊆Sα and let ft=gtt>0,limt⟶0+gtt=0. for g∈m. Then,(51)fA!Bu≤sec5α2In+fAuIn+fBu.Proof.
We have the following chain of inequalities:(52)fA!Bu≤secαℜfA!Buby Lemma2.4≤sec3αfℜA!Buby Lemma2.1≤sec5αfℜA!ℜBuby Lemma2.6≤sec5αfℜA!fℜBuby Theorem4in8≤sec5α2fℜA+fℜBuby1≤sec5α2In+fℜAuIn+fℜBuby11≤sec5α2In+ℜfAuIn+ℜfBuby Lemma2.1≤sec5α2In+fAuIn+fBuby Lemma2.2,which completes the proof.Remark 2.23.
In Theorem2.22, letting fX=X for any X∈Mnℂ, we have a better bound A!Bu≤sec3α/2In+AuIn+Bu.We close this paper with a determinant inequality.Theorem 2.24.
LetA,B∈Mnℂ be such that WA,WB⊆Sα. Then,(53)detA♯B≤sec3nα2ndetIn+A·detIn+B.Proof.
For positive semidefiniteS,T∈Mnℂ, let r=1 and k=n in (5); one can obtain(54)detS+T≤detIn+S·detIn+T.
Compute(55)detA♯B≤secnαdetℜA♯Bby Lemma2.6in15≤sec3nαdetℜA♯ℜBby Lemma2.6≤sec3nαdetℜA+ℜB2by1=sec3nα2ndetℜA+ℜB≤sec3nα2ndetIn+ℜA·detIn+ℜBby12≤sec3nα2ndetIn+A·detIn+Bby7.8.20in12,completing the proof.Remark 2.25.
We remark that Theorem2.24 is a refinement of Proposition 3.1 in [22].


## 3. Conclusion

In this paper, we utilize the properties of operator concave functions to obtain some singular value inequalities that generalize some existing results, and some unitarily invariant norm inequalities and determinant inequalities are also derived which are refinements of some known results. In the future, we plan to investigate some related singular value inequalities for sector matrices involving operator convex functions.



---

*Source: 4535343-2022-05-23.xml*