Abstract
In this paper, we investigate some operator inequalities for the p-Schatten norm and obtain some other versions of these inequalities when parameters taking values in different regions. Let such that . Then, for , and , . For , and , the inequalities are reversed. Moreover, we get some applications of our results.
1. Introduction
Let be the -algebra of all bounded linear operators acting on a complex separable Hilbert space H. denotes the absolute value of an operator . If is compact, let be the sequence of decreasingly ordered singular values of A. For , let , where is the nuclearity (or trace class) of integral operators. This defines the Schatten p-norm (quasi-norm, resp.) for (, resp.) on the setwhich is called the p-Schatten class of (see [1]). The Schatten p-norms are unitarily invariant and when , is called the trace norm of A.
There are some classical Clarkson’s inequalities for the Schatten p-norms of operators in (see [2, 3]). If A, , thenfor andfor . For , by (2) and (3), we havewhich 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 some p-Schatten inequalities have been given by Conde and Moslehian in [9] and Gao and Tian in [10].
Theorem 1 (see [9]). Let such that , then for , and ,
For , and , the inequalities are reversed.
Theorem 2 (see [9]). Let such that , then for , and ,
For , and , the inequality is reversed.
Theorem 3 (see [10]). Let such that . Then, for , and ,
For , and , the inequalities are reversed.
Theorem 4 (see [10]). Let such that . Then, for , and ,
For , and , the inequality is reversed.
In this paper, motivated by the above conclusions and techniques, we consider some other versions of p-Schatten norm inequalities when p, λ, and μ taking values in different regions.
2. Main Results
In this section, we consider the p-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. for , where is an n-tuple of non-negative numbers. Fact 2. for any with .
Lemma 1 (see [9]). Let such that , then
Lemma 2 (see [2, 11]). If for some and are positive, then for ,and for , the inequalities are reversed.
They are a refinement of Lemma 1 in [6]. A commutative version of the previous lemma for scalars is given in the following:
Let be an n-tuple of non-negative numbers, thenfor and
for .
Theorem 5. Let such that . Then, for , and ,
For , and , the inequalities are reversed.
Proof. Let . It follows from such thatApplying the well-known fact that for any with and Lemmas 1 and 2, we getUsing Lemma 2 and the convexity of the function on for , we obtainFor , and , we can prove the inequalities by the same reasoning.
Corollary 1. Let such that . Then,
Proof. Motivated by Theorem 5, let .
Corollary 2. Let such that . Then,
Proof. implies that . The statement is a consequence of Corollary 1.
Theorem 6. Let such that . Then, for , and ,
For , and , the inequality is reversed.
Proof. We suppose that , and . Then, by Lemma 2 we haveWhen , and , we can prove the inequality by the same reasoning.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (11701154) and the Natural Science Foundation of the Department of Education, Henan Province (19A110020 and 20A110020), the graduate education reform and quality improvement project, Henan Province.