Norm inequalities involving convex and concave functions of operators

Abstract

Let $A_1,\dots ,A_n$ be bounded linear operators on a complex separable Hilbert space $\mathbb{H}$ and let ${\alpha }_1,\dots ,{\alpha }_n$ be positive real numbers such that

${\sum }_{j=1}^n{\alpha }_j{A}_j=0$ and ${\sum }_{j=1}^n{\alpha }_j=1$. Among other results, it is shown that (a) If f is a non-negative function on $[0,\infty )$ such that $f(0)=0$ and $g(t)=f(\sqrt{t})$ is convex, then for every unitarily invariant norm,

$$\begin{array}{cc}\hfill \left|\left|\left|\sum _{j=1}^n{\alpha }_{j}f\left(\left|A_j\right|\right)\right|\right|\right|& \ge \left|\left|\left|f\left(\sqrt{\frac{{\alpha }_{\ell }}{1-{\alpha }_{\ell }}}\left|A_{\ell }\right|\right)\right.\right.\right.\hfill \\ \hfill & \left.\left.\left.+\sum _{j,k\in S_{\ell }}f\left(\sqrt{\frac{{\alpha }_j{\alpha }_k}{2\left(1-{\alpha }_{\ell }\right)}}\left|A_j-A_k\right|\right)\right|\right|\right|\hfill \end{array}$$

for $\ell =1,\dots ,n$.

(b) If f is a non-negative function on $[0,\infty )$ such that $g(t)=f(\sqrt{t})$ is concave, then for every unitarily invariant norm, $$\begin{array}{cc}\hfill \left|\left|\left|\sum _{j=1}^n{\alpha }_{j}f\left(\left|A_j\right|\right)\right|\right|\right|& \le \left|\left|\left|f\left(\sqrt{\frac{{\alpha }_{\ell }}{1-{\alpha }_{\ell }}}\left|A_{\ell }\right|\right)\right.\right.\right.\hfill \\ \hfill & \left.\left.\left.+\sum _{j,k\in S_{\ell }}f\left(\sqrt{\frac{{\alpha }_j{\alpha }_k}{2\left(1-{\alpha }_{\ell }\right)}}\left|A_j-A_k\right|\right)\right|\right|\right|\hfill \end{array}$$

for $\ell =1,\dots ,n$. Here $S_{\ell }=\{1,\dots ,n\}\backslash \{\ell \}.$

Keywords:

• Clarkson inequalities
• unitarily invariant norm
• Schatten p-norm
• convex function
• concave function

AMS Subject Classifications:

• 47A30
• 47B10
• 47B15
• 46B20

Notes

No potential conflict of interest was reported by the authors.