Abstract
We present some refinements of the Cauchy-Schwarz and Heinz inequalities for operators by utilizing a refinement of the Hermite-Hadamard inequality.
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1 Introduction
Let \(B(\mathcal{H})\) be the \(C^{*}\)-algebra of all bounded linear operators on a complex separable Hilbert space \(\mathcal{H}\). Let \(\Vert \cdot \Vert \) denote a unitarily invariant norm defined on a two-sided ideal \(\tau_{\Vert \cdot \Vert }\) that is included in \(C_{\infty}\) (the set of compact operators); it has the basic property \(\Vert UAV\Vert =\Vert A\Vert \) for every \(A\in\tau_{\Vert \cdot \Vert }\) and all unitary operators \(U, V\in B(\mathcal{H})\). For \(A\in B(\mathcal{H})\), \(\vert A\vert =(A^{*}A)^{\frac{1}{2}}\), where \(A^{*}\) is the conjugate operator of A.
Hiai and Zhan [1] proved that if \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive and \(X\in\tau_{\Vert \cdot \Vert }\), \(r>0\), then the function \(f(v)=\Vert \vert A^{v}XB^{1-v}\vert ^{r}\Vert \cdot \Vert \vert A^{1-v}XB^{v}\vert ^{r}\Vert \) is convex on the interval \([0,1]\) and attains its minimum at \(v=\frac{1}{2}\) maximum at \(v=0\) and \(v=1\). Consequently, it is decreasing on \([0,\frac{1}{2}]\) and increasing on \([\frac{1}{2},1]\); moreover, \(f(v)=f(1-v)\) for \(v\in [0,1]\). Thus for every unitarily invariant norm \(\Vert \cdot \Vert \), we have the operator inequality
for \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive, \(X\in\tau_{\Vert \cdot \Vert }\), \(r>0\), and \(v\in[0,1]\). The inequality \(\Vert \vert A^{\frac{1}{2}}XB^{\frac{1}{2}}\vert ^{r}\Vert ^{2} \leq \Vert \vert AX\vert ^{r}\Vert \cdot \Vert \vert XB\vert ^{r}\Vert \) is often called the Cauchy-Schwarz norm inequality due to Bhatia and Davis [2].
The Heinz means for operators are defined by \(g(v)=\Vert A^{v}XB^{1-v}+A^{1-v}XB^{v}\Vert \) for \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive, \(X\in\tau_{\Vert \cdot \Vert }\), and \(v\in[0,1]\), where \(\Vert \cdot \Vert \) is a unitarily invariant norm. Bhatia and Davis [3] obtained the inequality
for \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive, \(X\in\tau_{\Vert \cdot \Vert }\), and \(v\in[0,1]\). In fact, they proved that \(g(v)=\Vert A^{v}XB^{1-v}+A^{1-v}XB^{v}\Vert \) is a convex function of v on the interval \([0,1]\), attains its minimum at \(v=\frac{1}{2}\) and maximum at \(v=0\) and \(v=1\). Hence, it is decreasing on \([0,\frac{1}{2}]\) and increasing on \([\frac{1}{2},1]\); moreover, \(g(v)=g(1-v)\) for \(v\in [0,1]\). The second inequality in (2) is one of the most essential inequalities in operator theory, which is often called the Heinz inequality.
Recently, using the convexity of the function \(f(v)=\Vert \vert A^{v}XB^{1-v}\vert ^{r}\Vert \cdot \Vert \vert A^{1-v}XB^{v}\vert ^{r}\Vert \) (\(v\in[0,1]\)), Burqan [4], Theorem 2, obtained a refinement of inequality (1): Let \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive and \(X\in\tau_{\Vert \cdot \Vert }\). Then, for every unitarily invariant norm \(\Vert \cdot \Vert \),
where \(v\in[0,1]\) and \(r>0\).
Using the convexity of the function \(g(v)=\Vert A^{v}XB^{1-v}+A^{1-v}XB^{v}\Vert \) (\(v\in[0,1]\)), Feng [5], Theorem 1, obtained a refinement of the first inequality in (2): Let \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive and \(X\in\tau_{\Vert \cdot \Vert }\). Then, for every unitarily invariant norm \(\Vert \cdot \Vert \),
where \(v\in[0,1]\).
In this note, we present some refinements of norm inequalities (1), (2), (3), and (4), also using the convexity of the functions f and g.
2 Main results
In this section, we mainly present some refinements of the Cauchy-Schwarz and Heinz inequalities for operators. To achieve our goal, we need some preparations.
Let \(f:I\rightarrow R\) be a real-valued convex function on the interval \(I\subseteq R\). Let \(a, b\in I\) with \(a< b\). The inequality
is well known as the Hermite-Hadamard inequality, which includes a basic property of convex functions.
Now, we present some lemmas.
Lemma 2.1
Let f be a real-valued convex function on the interval \([a,b]\). Then
Proof
Thanks to the Hermite-Hadamard’s inequality (5), we obtain
and
Similarly, due to the convexity of f on the interval \([\frac{a+b}{2},b]\), we have
Hence, by inequalities (9) and (10) we deduce
This completes the proof. □
Remark 2.2
Let f be a real-valued convex function on the interval \([a,b]\). By the convexity of f on the interval \([a,\frac{a+b}{2}]\) we get
Similarly, due to the convexity of f on the interval \([\frac{a+b}{2},b]\), we have
Hence, inequality (6) is a refinement of the inequality
obtained by Burqan [4], Lemma 1.
Lemma 2.3
Let f be a real-valued convex function on the interval \([a,b]\). Then
Proof
By the Hermite-Hadamard’s inequality (5) we obtain
This completes the proof. □
Remark 2.4
Let f be a real-valued convex function on the interval \([a,b]\). Thanks to the convexity of f on the interval \([a,\frac{a+b}{2}]\) and \([\frac{a+b}{2},b]\), respectively, we obtain
and
Hence, combining inequalities (14) and (15), we get
Hence, inequality (13) is a refinement of the inequality
obtained by Feng [5], Lemma 2.
Combining Lemmas 2.1 and 2.3, we obtain the following lemma.
Lemma 2.5
Let f be a real-valued convex function on the interval \([a,b]\). Then
Remark 2.6
By inequalities (11), (12), and (16) it is easy to see that inequality (17) is a refinement of inequality (5).
Based on Lemma 2.5, we obtain the following theorems.
Theorem 2.7
Let \(\Vert \cdot \Vert \) be a unitarily invariant norm, and \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive and \(X\in\tau_{\Vert \cdot \Vert }\). Then
where \(v\in[0,1]\setminus\{\frac{1}{2}\}\) and \(r>0\).
Proof
We first consider the case \(0\leq v<\frac{1}{2}\). Applying Lemma 2.5 to the function \(f(x)=\Vert \vert A^{x}XB^{1-x}\vert ^{r}\Vert \cdot \Vert \vert A^{1-x}XB^{x}\vert ^{r}\Vert \) on the interval \([v, 1-v]\) and \(f(x)=f(1-x)\) for \(x\in[0,1]\), we obtain
Thus, we get inequality (18) for \(0\leq v<\frac{1}{2}\).
Similarly, when \(\frac{1}{2}< v\leq1\), applying Lemma 2.5 to the function f on the interval \([1-v,v]\) and \(f(x)=f(1-x)\) for \(x\in[0,1]\), we obtain
Hence, we get inequality (18) for \(\frac{1}{2}< v\leq1\).
This completes the proof. □
The next theorem is a refinement of Heinz’s norm inequalities.
Theorem 2.8
Let \(\Vert \cdot \Vert \) be a unitarily invariant norm, and \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive and \(X\in\tau_{\Vert \cdot \Vert }\). Then
where \(v\in[0,1]\setminus\{\frac{1}{2}\}\).
Proof
Replacing \(f(x)=\Vert \vert A^{x}XB^{1-x}\vert ^{r}\Vert \cdot \Vert \vert A^{1-x}XB^{x}\vert ^{r}\Vert \) by \(g(x)=\Vert A^{x}XB^{1-x}+A^{1-x}XB^{x}\Vert \) in the proof of Theorem 2.7, we get the desired result. □
Remark 2.9
Putting \(f(x)=\Vert \vert A^{x}XB^{1-x}\vert ^{r}\Vert \cdot \Vert \vert A^{1-x}XB^{x}\vert ^{r}\Vert \) (\(x\in[0,1]\)), where \(\Vert \cdot \Vert \) is a unitarily invariant norm, \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive, \(X\in\tau_{\Vert \cdot \Vert }\), and \(r>0\). By the convexity of f we have \(f (\frac{2v+1}{4} )\leq\frac{1}{2} \{f (\frac{6v+1}{8} )+ f (\frac{2v+3}{8} ) \}\) and \(\frac{1}{4} \{f(v)+2f (\frac{2v+1}{4} )+f (\frac{1}{2} ) \} \leq\frac{1}{2}\{f(v)+f(\frac{1}{2})\}\) for \(v\in[0,1]\setminus\{\frac{1}{2}\}\). So, norm inequality (18) is a refinement of inequality (3) obtained by Burqan [4], Theorem 2.
Remark 2.10
Putting \(g(x)=\Vert A^{x}XB^{1-x}+A^{1-x}XB^{x}\Vert \) \((x\in[0,1])\), where \(\Vert \cdot \Vert \) is a unitarily invariant norm, and \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive and \(X\in\tau_{\Vert \cdot \Vert }\). By the convexity of g we have \(g (\frac{2v+1}{4} )\leq\frac{1}{2} \{g (\frac{6v+1}{8} )+ g (\frac{2v+3}{8} ) \}\) and \(\frac{1}{4} \{g(v)+2g (\frac{2v+1}{4} )+g (\frac{1}{2} ) \} \leq\frac{1}{2}\{g(v)+g(\frac{1}{2})\}\) for \(v\in[0,1]\setminus\{\frac{1}{2}\}\). Hence, norm inequality (19) is a refinement of inequality (4) obtained by Feng [5], Theorem 1.
For \(0< v<1\), applying Lemma 2.5 to the convex functions \(f(x)=\Vert \vert A^{x}XB^{1-x}\vert ^{r}\Vert \cdot \Vert \vert A^{1-x}XB^{x}\vert ^{r}\Vert \) and \(g(x)=\Vert A^{x}XB^{1-x}+A^{1-x}XB^{x}\Vert \) on the interval \([0,v]\) when \(0< v\leq\frac{1}{2}\) and on the interval \([v,1]\) when \(\frac{1}{2}\leq v<1\), respectively, we obtain the following theorems, where \(\Vert \cdot \Vert \) is a unitarily invariant norm, \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive and \(X\in\tau_{\Vert \cdot \Vert }\), and \(r>0\).
Theorem 2.11
Let \(\Vert \cdot \Vert \) be a unitarily invariant norm, and \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive and \(X\in\tau_{\Vert \cdot \Vert }\). Then
(i) for \(0< v\leq\frac{1}{2}\),
(ii) for \(\frac{1}{2}< v<1\),
where \(r>0\).
Theorem 2.12
Let \(\Vert \cdot \Vert \) be a unitarily invariant norm, and \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive and \(X\in\tau_{\Vert \cdot \Vert }\). Then
(i) for \(0< v\leq\frac{1}{2}\),
(ii) for \(\frac{1}{2}< v\leq 1\),
Remark 2.13
Putting \(f(x)=\Vert \vert A^{x}XB^{1-x}\vert ^{r}\Vert \cdot \Vert \vert A^{1-x}XB^{x}\vert ^{r}\Vert \), where \(\Vert \cdot \Vert \) is a unitarily invariant norm, \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive and \(X\in\tau_{\Vert \cdot \Vert }\), and \(r>0\). We first consider the case \(0< v\leq\frac{1}{2}\). Since f is decreasing on the interval \([0,\frac{1}{2}]\), we have
Using inequalities (12) and (16) for f on the interval \([0,v]\), respectively, we get
and
By the convexity of f on the interval \([0,v]\) we have
Noting that
by (i) of Theorem 2.11 we have
Combining inequalities (20), (21), (22), (23), (24), and (25), we have
Similarly, when \(\frac{1}{2}< v<1\), since \(f(v)\) is increasing on the interval \([\frac{1}{2},1]\), we have \(f(v)\leq f(\frac{1+v}{2})\). By the same process as before, we also have
Thus, inequalities (26) and (27) are refinements of the second inequality in (1) and also a refinement of Theorem 4 of Burqan [4].
Remark 2.14
Putting \(g(x)=\Vert A^{x}XB^{1-x}+A^{1-x}XB^{x}\Vert \), where \(\Vert \cdot \Vert \) is a unitarily invariant norm, and \(A, B, X\in B(\mathcal{H})\) with \(A, B\) positive and \(X\in\tau_{\Vert \cdot \Vert }\). Replacing f by g in Remark 2.13, we obtain
where \(0< v\leq\frac{1}{2}\).
Similarly, when \(\frac{1}{2}< v<1\), we have
Therefore, inequalities (28) and (29) are refinements of the second inequality in (2) and also a refinement of Theorem 3 of Feng [5].
References
Hiai, F, Zhan, X: Inequalities involving unitarily invariant norms and operator monotone functions. Linear Algebra Appl. 341, 151-169 (2002)
Bhatia, R, Davis, C: A Cauchy-Schwarz inequality for operators with application. Linear Algebra Appl. 223/224, 119-129 (1995)
Bhatia, R, Davis, C: More matrix forms of the arithmetic-geometric mean inequality. SIAM J. Matrix Anal. Appl. 14, 132-136 (1993)
Burqan, A: Improved Cauchy-Schwarz norm inequality for operators. J. Math. Inequal. 10(1), 205-211 (2016)
Feng, Y: Refinements of the Heinz inequalities. J. Inequal. Appl. 2012, 18 (2012)
Acknowledgements
The authors would like to express their thanks to Editor and reviewers for their valuable comments and suggestions on our manuscript. This work is supported by the National Natural Science Foundations of China (Grant No. 11161040).
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Zhao, J., Xie, H., Wu, J. et al. Norm inequalities for operators related to the Cauchy-Schwarz and Heinz inequalities. J Inequal Appl 2016, 274 (2016). https://doi.org/10.1186/s13660-016-1216-8
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DOI: https://doi.org/10.1186/s13660-016-1216-8