# Versions of Inequalities Related top-Schatten Norm

Authors: Fugen Gao; Xiaoxiao Liu; Mengyu Tian
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we investigate some operator inequalities for thep-Schatten norm and obtain some other versions of these inequalities when parameters taking values in different regions. Let A1,…,An,B1,…,Bn∈BpH such that ∑i,j=1nAi∗Bj=0. Then, for p≥2,p≥λ>0, and 0<μ≤2, n21/p−1/λ∑i,j=1nAi±Bjpλ1/λ≤n1/2∑i=1nAi2+∑i=1nBi2p/21/2≤21/2−μ/4n1−μ/4∑i=1nAip4/μ+∑i=1nBip4/μμ/4. For 0<p≤2,p≤λ, and μ≥2, the inequalities are reversed. Moreover, we get some applications of our results.


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## Body

## 1. Introduction

LetBH be the C∗-algebra of all bounded linear operators acting on a complex separable Hilbert space H. A=A∗A1/2 denotes the absolute value of an operator A∈BH. If A∈BH is compact, let sjAj=1∞ be the sequence of decreasingly ordered singular values of A. For 0<p<∞, let Ap=trAp1/p=∑j=1∞sjpA1/p, where tr is the nuclearity (or trace class) of integral operators. This defines the Schatten p-norm (quasi-norm, resp.) for 1≤p<∞ (0<p<1, resp.) on the set(1)BpH=X∈BH:Xp<∞,which is called the p-Schatten class of BH (see [1]). The Schatten p-norms are unitarily invariant and when p=1, A1=trA is called the trace norm of A.There are some classical Clarkson’s inequalities for the Schattenp-norms of operators in BpH (see [2, 3]). If A, B∈BpH, then(2)2p−1App+Bpp≤A−Bpp+A+Bpp≤2App+Bpp,for 0<p≤2 and(3)2App+Bpp≤A−Bpp+A+Bpp≤2p−1App+Bpp,for 2≤p<∞. For p=2, by (2) and (3), we have(4)A−B22+A+B22=2A22+B22,which is called the parallelogram law.Bhatia et al. have obtained some generalizations of (2) to n-tuples of operators and many different conclusions by using various methods such as complex interpolation method and concavity and convexity of certain functions (see [3–8]).Recently, some other versions of somep-Schatten inequalities have been given by Conde and Moslehian in [9] and Gao and Tian in [10].Theorem 1 (see [9]).
LetA1,…,An,B1,…,Bn∈BpH such that ∑i,j=1nAi∗Bj=0, then for 0<p≤2,p≤λ, and 0<μ≤2,(5)21/2−1/μn1−1/μ∑i=1nAipμ+∑i=1nBipμ1/μ≤n1/2∑i=1nAip2+∑i=1nBip21/2≤n21/p−1/λ∑i,j=1nAi±Bjpλ1/λ.For2≤p,0<λ≤p, and 2≤μ, the inequalities are reversed.Theorem 2 (see [9]).
LetA1,…,An,B1,…,Bn∈BpH such that ∑i,j=1nAi∗Bj=0, then for 0<p≤2,p≤λ, and 0<μ≤2,(6)n1n2∑i,j=1nAi±Bjpμ1/μ≤n1/2+1/p−1/λ∑i=1nAi2+Bi21/2pλ1/λ.For2≤p,0<λ≤p, and 2≤μ, the inequality is reversed.Theorem 3 (see [10]).
LetA1,…,An,B1,…,Bn∈BpH such that ∑i,j=1nAi∗Bj=0. Then, for 0<p≤2,p≥λ>0, and μ≥2,(7)n21/p−1/λ∑i,j=1nAi±Bjpλ1/λ≤n2/p−1/2∑i=1nAipp+∑i=1nBipp1/p≤21/p−2/pμn3/p−2/pμ−1/2∑i=1nAippμ/2+∑i=1nBippμ/22/pμ.Forp≥2,p≤λ, and 0<μ≤2, the inequalities are reversed.Theorem 4 (see [10]).
LetA1,…,An,B1,…,Bn∈BpH such that ∑i,j=1nAi∗Bj=0. Then, for 0<p≤2,p≥λ>0, and μ≥2,(8)n1n2∑i,j=1nAi±Bjpμ1/μ≥n2−2/p−1/λ∑i=1nAi2+Bi21/2pλ1/λ.Forp≥2,p≤λ, and 0<μ≤2, the inequality is reversed.In this paper, motivated by the above conclusions and techniques, we consider some other versions ofp-Schatten norm inequalities when p, λ, and μ taking values in different regions.


## 2. Main Results

In this section, we consider thep-Schatten norm inequalities of (5) and (6) when parameters taking values in different regions. We start our work with the following lemmas that we will use along the paper.Fact 1.Msx¯≤Ms′x¯ for 0<s≤s′, where Msx¯=1/n∑i=1nxis1/s,x¯=x1,…,xn is an n-tuple of non-negative numbers.Fact 2.Tp2=T2p/2 for any T∈BpH with p>0.Lemma 1 (see [9]).
LetA1,…,An,B1,…,Bn∈BH such that ∑i,j=1nAi∗Bj=0, then(9)∑i,j=1nAi±Bj2=∑i,j=1nAi2+Bj2±∑i,j=1nAi∗Bj+Bj∗Ai=∑i,j=1nAi2+Bj2.Lemma 2 (see [2, 11]).
IfA1,…,An∈BpH for some p>0 and A1,…,An are positive, then for 0<p≤1,(10)np−1∑i=1nAipp≤∑i=1nAipp≤∑i=1nAipp≤∑i=1nAipp,and for 1≤p<∞, the inequalities are reversed.They are a refinement of Lemma1 in [6]. A commutative version of the previous lemma for scalars is given in the following:Letx¯=x1,…,xn be an n-tuple of non-negative numbers, then(11)np−1∑i=1nxip≤∑i=1nxip≤∑i=1nxip,for 0<p≤1 and(12)∑i=1nxip≤∑i=1nxip≤np−1∑i=1nxip,for1≤p<∞.Theorem 5.
LetA1,…,An,B1,…,Bn∈BpH such that ∑i,j=1nAi∗Bj=0. Then, for p≥2,p≥λ>0, and 0<μ≤2,(13)n21/p−1/λ∑i,j=1nAi±Bjpλ1/λ≤n1/2∑i=1nAi2+∑i=1nBi2p/21/2≤21/2−μ/4n1−μ/4∑i=1nAip4/μ+∑i=1nBip4/μμ/4.For0<p≤2,p≤λ, and μ≥2, the inequalities are reversed.Proof.
Letp≥2,p≥λ>0,0<μ≤2. It follows from Mλx¯≤Mpx¯ such that(14)n21/p−1/λ∑i,j=1nAi±Bjpλ1/λ=n2/p1n2∑i,j=1nAi±Bjpλ1/λ≤∑i,j=1nAi±Bjpp1/p.
Applying the well-known fact thatTp2=T2p/2 for any T∈BpH with p>0 and Lemmas 1 and 2, we get(15)∑i,j=1nAi±Bjpp1/p=∑i,j=1nAi±Bj2p/2p/21/p≤∑i,j=1nAi±Bj2p/2p/21/p=∑i,j=1nAi2+Bj2p/21/2=n1/2∑i=1nAi2+∑i=1nBi2p/21/2.
Using Lemma2 and the convexity of the function fx=xα on 0,+∞ for 1≤α<∞, we obtain(16)n1/2∑i=1nAi2+∑i=1nBi2p/21/2=n1/2∑i=1nAi2+∑i=1nBi2p/22/μμ/4≤n1/2∑i=1nAi2p/2+∑i=1nBi2p/22/μμ/4=n1/22n2/μ12n∑i=1nAi2p/2+∑i=1nBi2p/22/μμ/4≤21/2n12n∑i=1nAi2p/22/μ+∑i=1nBi2p/22/μμ/4=21/2−μ/4n1−μ/4∑i=1nAip4/μ+∑i=1nBip4/μμ/4.
For0<p≤2,p≤λ, and μ≥2, we can prove the inequalities by the same reasoning.Corollary 1.
LetA1,…,An,B1,…,Bn∈BpH such that ∑i,j=1nAi∗Bj=0. Then,(17)∑i,j=1nAi±Bj22=n∑i=1nAi22+∑i=1nBi22.Proof.
Motivated by Theorem5, let λ=μ=p=2.Corollary 2.
LetA1,…,An∈BpH such that ∑i=1nAi=0. Then,(18)∑i,j=1nAi±Aj22=2n∑i=1nAi22.Proof.
∑
i
=
1
n
A
i
=
0 implies that ∑i,j=1nAi∗Aj=0. The statement is a consequence of Corollary 1.Theorem 6.
LetA1,…,An,B1,…,Bn∈BpH such that ∑i,j=1nAi∗Bj=0. Then, for p≥2,p≥λ>0, and 0<μ≤2,(19)n1n2∑i,j=1nAi±Bjpμ1/μ≤n2+1/λ−4/p∑i=1nAi2+Bi21/2pλ1/λ.For0<p≤2,p≤λ, and μ≥2, the inequality is reversed.Proof.
We suppose thatp≥2,p≥λ>0, and 0<μ≤2. Then, by Lemma 2 we have(20)n1n2∑i,j=1nAi±Bjpμ1/μ=n1n2∑i,j=1nAi±Bj2p/2μ/21/μ≤n1n2∑i,j=1nAi±Bj2p/2μ/21/μ=∑i,j=1nAi±Bj2p/21/2≤n2p/2−1∑i,j=1nAi±Bj2p/2p/21/p=n1−2/p∑i,j=1nAi±Bj2p/2p/21/p≤n1−2/p∑i,j=1nAi±Bj2p/2p/21/p=n3/2−2/p∑i=1nAi2+Bi2p/2p/21/p≤n3/2−2/pnp/2−1∑i=1nAi2+Bi2p/2p/21/p=n2−3/p∑i=1nAi2+Bi21/2pp1/p≤n2−3/pnp/λ−1∑i=1nAi2+Bi21/2pλp/λ1/p=n2+1/λ−4/p∑i=1nAi2+Bi21/2pλ1/λ.
When0<p≤2,p≤λ, and μ≥2, we can prove the inequality by the same reasoning.



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*Source: 9365964-2020-02-24.xml*